[ { "id": "authors:ywbjj-n1p25", "collection": "authors", "collection_id": "ywbjj-n1p25", "cite_using_url": "https://authors.library.caltech.edu/records/ywbjj-n1p25", "type": "article", "title": "Borel graphable equivalence relations", "author": [ { "family_name": "Arant", "given_name": "Tyler" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Lutz", "given_name": "Patrick" } ], "abstract": "

This paper is devoted to the study of analytic equivalence relations which are Borel graphable, i.e. which can be realized as the connectedness relation of a Borel graph. Our main focus is the question of which analytic equivalence relations are Borel graphable. First, we study an equivalence relation arising from the theory of countable admissible ordinals and show that it is Borel graphable if and only if there is a non-constructible real. As a corollary of the proof, we construct an analytic equivalence relation which is (provably in ZFC) not Borel graphable and an effectively analytic equivalence relation which is Borel graphable but not effectively Borel graphable. Next, we study analytic equivalence relations given by the isomorphism relation for some class of countable structures. We show that all such equivalence relations are Borel graphable, which implies that for every Borel action of S ∞, the associated orbit equivalence relation is Borel graphable. This leads us to study the class of Polish groups whose Borel actions always give rise to Borel graphable orbit equivalence relations; we refer to such groups as graphic groups. We show that besides S ∞, the class of graphic groups includes all connected Polish groups and is closed under countable products. We finish by studying structural properties of the class of Borel graphable analytic equivalence relations and by considering two variations on Borel graphability: a generalization with hypergraphs instead of graphs and an analogue of Borel graphability in the setting of computably enumerable equivalence relations.

", "doi": "10.1016/j.aim.2025.110765", "issn": "0001-8708", "publisher": "Elsevier", "publication": "Advances in Mathematics", "publication_date": "2026-03", "volume": "487", "pages": "110765" }, { "id": "authors:25tmg-55397", "collection": "authors", "collection_id": "25tmg-55397", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20230125-514112400.2", "type": "article", "title": "Lifts of Borel actions on quotient spaces", "author": [ { "family_name": "Frisch", "given_name": "Joshua R.", "clpid": "Frisch-Joshua-R" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Shinko", "given_name": "Forte", "orcid": "0000-0001-8142-1509", "clpid": "Shinko-Forte" } ], "abstract": "Given a countable Borel equivalence relation E and a countable group G, we study the problem of when a Borel action of G on X/E can be lifted to a Borel action of G on X.", "doi": "10.1007/s11856-022-2434-z", "issn": "0021-2172", "publisher": "Springer", "publication": "Israel Journal of Mathematics", "publication_date": "2022-12", "series_number": "2", "volume": "251", "issue": "2", "pages": "379-421" }, { "id": "authors:yk4b8-2q134", "collection": "authors", "collection_id": "yk4b8-2q134", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20220107-488030400", "type": "article", "title": "On Polish groups admitting non-essentially countable actions", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Malicki", "given_name": "Maciej", "clpid": "Malicki-Maciej" }, { "family_name": "Panagiotopoulos", "given_name": "Aristotelis", "orcid": "0000-0002-7695-4842", "clpid": "Panagiotopoulos-Aristotelis" }, { "family_name": "Zielinski", "given_name": "Joseph", "clpid": "Zielinski-Joseph" } ], "abstract": "It is a long-standing open question whether every Polish group that is not locally compact admits a Borel action on a standard Borel space whose associated orbit equivalence relation is not essentially countable. We answer this question positively for the class of all Polish groups that embed in the isometry group of a locally compact metric space. This class contains all non-archimedean Polish groups, for which we provide an alternative proof based on a new criterion for non-essential countability. Finally, we provide the following variant of a theorem of Solecki: every infinite-dimensional Banach space has a continuous action whose orbit equivalence relation is Borel but not essentially countable.", "doi": "10.1017/etds.2020.133", "issn": "0143-3857", "publisher": "Cambridge University Press", "publication": "Ergodic Theory and Dynamical Systems", "publication_date": "2022-01", "series_number": "1", "volume": "42", "issue": "1", "pages": "180-194" }, { "id": "authors:gdfsz-5f343", "collection": "authors", "collection_id": "gdfsz-5f343", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-090422831", "type": "article", "title": "Weak containment of measure-preserving group actions", "author": [ { "family_name": "Burton", "given_name": "Peter J.", "clpid": "Burton-P-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "This paper concerns the study of the global structure of measure-preserving actions of countable groups on standard probability spaces. Weak containment is a hierarchical notion of complexity of such actions, motivated by an analogous concept in the theory of unitary representations. This concept gives rise to an associated notion of equivalence of actions, called weak equivalence, which is much coarser than the notion of isomorphism (conjugacy). It is well understood now that, in general, isomorphism is a very complex notion, a fact which manifests itself, for example, in the lack of any reasonable structure in the space of actions modulo isomorphism. On the other hand, the space of weak equivalence classes is quite well behaved. Another interesting fact that relates to the study of weak containment is that many important parameters associated with actions, such as the type, cost, and combinatorial parameters, turn out to be invariants of weak equivalence and in fact exhibit desirable monotonicity properties with respect to the pre-order of weak containment, a fact that can be useful in certain applications. There has been quite a lot of activity in this area in the last few years, and our goal in this paper is to provide a survey of this work.", "doi": "10.1017/etds.2019.26", "issn": "0143-3857", "publisher": "Cambridge University Press", "publication": "Ergodic Theory and Dynamical Systems", "publication_date": "2020-10", "series_number": "10", "volume": "40", "issue": "10", "pages": "2681-2733" }, { "id": "authors:xx7r1-n0q03", "collection": "authors", "collection_id": "xx7r1-n0q03", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20210707-142840752", "type": "book_section", "title": "Quasi-invariant measures for continuous group actions", "book_title": "Trends in Set Theory", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Coskey", "given_name": "Samuel", "clpid": "Coskey-Samuel" }, { "family_name": "Sargsyan", "given_name": "Grigor", "clpid": "Sargsyan-Grigor" } ], "abstract": "The class of ergodic, invariant probability Borel measure for the shift action of a countable group is a G_\u03b4 set in the compact, metrizable space of probability Borel measures. We study in this paper the descriptive complexity of the class of ergodic, quasi-invariant probability Borel measures and show that for any infinite countable group \u0393 it is \u03a0\u2070\u2083-hard, for the group Z it is \u03a0\u2070\u2083-complete, while for the free group F_\u221e with infinite, countably many generators it is \u03a0\u2070_\u03b1-complete, for some ordinal \u03b1 with 3 \u2264 \u03b1 \u2264 \u03c9 +2. The exact value of this ordinal is unknown.", "doi": "10.1090/conm/752/15132", "isbn": "978-1-4704-4332-0", "publisher": "American Mathematical Society", "place_of_publication": "Providence, RI", "publication_date": "2020", "pages": "113-119" }, { "id": "authors:pxrsq-vyd35", "collection": "authors", "collection_id": "pxrsq-vyd35", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20191205-100802061", "type": "article", "title": "Co-induction and invariant random subgroups", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Quorning", "given_name": "Vibeke", "clpid": "Quorning-V" } ], "abstract": "In this paper we develop a co-induction operation which transforms an invariant random subgroup of a group into an invariant random subgroup of a larger group.\nWe use this operation to construct new continuum size families of non-atomic, weakly mixing invariant random subgroups of certain classes of wreath products, HNN-extensions and free products with amalgamation. By use of small cancellation theory, we also construct a new continuum size family of non-atomic invariant random subgroups of F\u2082 which are all invariant and weakly mixing with respect to the action of Aut(F\u2082).\nMoreover, for amenable groups \u0393 \u2264 \u0394, we obtain that the standard co-induction operation from the space of weak equivalence classes of \u0393\u0393 to the space of weak equivalence classes of \u0394 is continuous if and only if [\u0394:\u0393] < \u221e or core \u0394(\u0393) is trivial. For general groups we obtain that the co-induction operation is not continuous when [\u0394:\u0393] = \u221e. This answers a question raised by Burton and Kechris in [17]. Independently such an answer was also obtained, using a different method, by Bernshteyn in [8].", "doi": "10.4171/GGD/517", "issn": "1661-7207", "publisher": "European Mathematical Society", "publication": "Groups, Geometry, and Dynamics", "publication_date": "2019-05-07", "series_number": "4", "volume": "13", "issue": "4", "pages": "1151-1193" }, { "id": "authors:ah2hj-4fp82", "collection": "authors", "collection_id": "ah2hj-4fp82", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-083759861", "type": "article", "title": "The complexity of topological group isomorphism", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Nies", "given_name": "Andr\u00e9", "clpid": "Nies-A" }, { "family_name": "Tent", "given_name": "Katrin", "clpid": "Tent-K" } ], "abstract": "We study the complexity of the topological isomorphism relation for various classes of closed subgroups of the group of permutations of the natural numbers. We use the setting of Borel reducibility between equivalence relations on Borel spaces. For profinite, locally compact, and Roelcke precompact groups, we show that the complexity is the same as the one of countable graph isomorphism. For oligomorphic groups, we merely establish this as an upper bound.", "doi": "10.1017/jsl.2018.25", "issn": "0022-4812", "publisher": "Cambridge University Press", "publication": "Journal of Symbolic Logic", "publication_date": "2018-09", "series_number": "3", "volume": "83", "issue": "3", "pages": "1190-1203" }, { "id": "authors:gsa52-xsw68", "collection": "authors", "collection_id": "gsa52-xsw68", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-102415675", "type": "article", "title": "Structurable equivalence relations", "author": [ { "family_name": "Chen", "given_name": "Ruiyuan", "clpid": "Chen-Ruiyuan" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "For a class K of countable relational structures, a countable Borel equivalence relation E is said to be K-structurable if there is a Borel way to put a structure in K on each E-equivalence class. We study in this paper the global structure of the classes of K-structurable equivalence relations for various K. We show that K-structurability interacts well with several kinds of Borel homomorphisms and reductions commonly used in\nthe classification of countable Borel equivalence relations. We consider the poset of classes\nof K-structurable equivalence relations for various K, under inclusion, and show that it is a distributive lattice; this implies that the Borel reducibility preordering among countable Borel equivalence relations contains a large sublattice. Finally, we consider the effect on K-structurability of various model-theoretic properties of K. In particular, we characterize the K such that every K-structurable equivalence relation is smooth, answering a question of Marks.", "doi": "10.4064/fm428-7-2017", "issn": "0016-2736", "publisher": "Institute of Mathematics, Polish Academy of Sciences", "publication": "Fundamenta Mathematicae", "publication_date": "2018-03-01", "volume": "242", "pages": "109-185" }, { "id": "authors:yzec3-5ya52", "collection": "authors", "collection_id": "yzec3-5ya52", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170712-091809776", "type": "article", "title": "Invariant random subgroups and action versus representation maximality", "author": [ { "family_name": "Burton", "given_name": "Peter J.", "clpid": "Burton-P-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We show that weak containment of free ergodic measure-preserving actions of F\u221e is not equivalent to weak containment of the corresponding Koopman representations. This result is based on the construction of an invariant random subgroup of F\u221e which is supported on the maximal actions.", "doi": "10.1090/proc/13591", "issn": "0002-9939", "publisher": "American Mathematical Society", "publication": "Proceedings of the American Mathematical Society", "publication_date": "2017-09", "series_number": "9", "volume": "145", "issue": "9", "pages": "3961-3971" }, { "id": "authors:4c0nr-xb414", "collection": "authors", "collection_id": "4c0nr-xb414", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20170630-111205546", "type": "book_section", "title": "Ramsey properties of finite measure algebras and topological dynamics of the group of measure preserving automorphisms: Some results and an open problem", "book_title": "Foundations of Mathematics: Logic at Harvard: Essays in Honor of W. Hugh Woodin's 60th Birthday", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Soki\u0107", "given_name": "M.", "clpid": "Soki\u0107-M" }, { "family_name": "Todorcevic", "given_name": "S.", "clpid": "Todor\u010devi\u0107-S" } ], "contributor": [ { "family_name": "Caicedo", "given_name": "Andr\u00e9s Eduardo", "clpid": "Caicedo-A-E" }, { "family_name": "Cummings", "given_name": "James", "clpid": "Cummings-J" }, { "family_name": "Koellner", "given_name": "Peter", "clpid": "Koellner-P" }, { "family_name": "Larson", "given_name": "Paul B.", "clpid": "Larson-P-B" } ], "abstract": "We study in this paper ordered finite measure algebras from the point of view of Fra\u00efss\u00e9 and Ramsey theory. We also propose an open problem, which is a homogeneous version of the Dual Ramsey Theorem of Graham-Rothschild, and derive consequences of a positive answer to the study of the\ntopological dynamics of the automorphism group of a standard probability space and also the group of measure preserving homeomorphisms of the Cantor space.", "doi": "10.1090/conm/690", "isbn": "978-1-4704-2256-1", "publisher": "American Mathematical Society", "place_of_publication": "Providence, RI", "publication_date": "2017", "pages": "69-85" }, { "id": "authors:9sngr-rxs67", "collection": "authors", "collection_id": "9sngr-rxs67", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20161130-102008360", "type": "article", "title": "Borel equivalence relations and cardinal algebras", "author": [ { "family_name": "Macdonald", "given_name": "Henry L.", "clpid": "Macdonald-H-L" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We show that Tarski's concept of cardinal algebra appears naturally in the context of the current theory of Borel equivalence relations. As a result one can apply Tarski's theory to discover a number of interesting laws governing the structure of Borel equivalence relations, which, in retrospect rather surprisingly, have not been realized before.", "doi": "10.4064/fm242-4-2016", "issn": "0016-2736", "publisher": "Institute of Mathematics, Polish Academy of Sciences", "publication": "Fundamenta Mathematicae", "publication_date": "2016-06-17", "volume": "235", "pages": "183-198" }, { "id": "authors:awjpx-r2a62", "collection": "authors", "collection_id": "awjpx-r2a62", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180806-100432570", "type": "book_section", "title": "Introduction to Q-theory", "book_title": "Ordinal Definability and Recursion Theory: The Cabal Seminar", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Martin", "given_name": "Donald A.", "clpid": "Martin-D-A" }, { "family_name": "Solovay", "given_name": "Robert M.", "clpid": "Solovay-R-M" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "abstract": "[no abstract]", "doi": "10.1017/CBO9781139519694.006", "isbn": "9781139519694", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2016", "pages": "126-199" }, { "id": "authors:j19aa-kf757", "collection": "authors", "collection_id": "j19aa-kf757", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180806-093515411", "type": "book_section", "title": "On Spector classes", "book_title": "Ordinal Definability and Recursion Theory: The Cabal Seminar", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "abstract": "[no abstract]", "doi": "10.1017/CBO9781139519694.013", "isbn": "9781139519694", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2016", "pages": "390-423" }, { "id": "authors:6b3nx-xsc69", "collection": "authors", "collection_id": "6b3nx-xsc69", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180806-092249394", "type": "book_section", "title": "On the theory of \u03a013 sets of reals, II", "book_title": "Ordinal Definability and Recursion Theory: The Cabal Seminar", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Martin", "given_name": "Donald A.", "clpid": "Martin-D-A" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "abstract": "[no abstract]", "doi": "10.1017/CBO9781139519694.007", "isbn": "9781139519694", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2016", "pages": "200-219" }, { "id": "authors:4bfne-24b32", "collection": "authors", "collection_id": "4bfne-24b32", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180806-100030179", "type": "book_section", "title": "Preface", "book_title": "Ordinal Definability and Recursion Theory: The Cabal Seminar", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "abstract": "This book continues the series of volumes containing reprints of the papers in the original Cabal Seminar volumes of the Springer Lecture Notes in Mathematics series [Cabal i, Cabal ii, Cabal iii, Cabal iv], unpublished material, and new papers. The first volume, [Cabal I], contained papers on games, scales and Suslin cardinals. The second volume, [Cabal II], contained papers on Wadge degrees and pointclasses and projective ordinals. In this volume, we continue with Parts V and VI of the project: Ordinal definability in models of determinacy and Recursion theory. As in our first two volumes, each of the parts contains an introductory survey (written by John Steel for Part V and by Leo Harrington and Ted Slaman for Part VI) putting the papers into a present-day context.\n\nIn addition to the reprinted papers, this volume contains a paper by Kechris and Martin (On the theory of \u03a0^1_3 sets of reals, II) that dates back to the period of the original Cabal publications but was not included in the old volumes. Neeman contributed a new paper, An inner models proof of the Kechris\u2013Martin theorem, related to this paper. Steel and Woodin contributed two new papers (A theorem of Woodin on mouse sets, authored by Steel, and HOD as a core model, jointly) with recent results that fit well with the topics of Part V. There is also a new paper by Marks, Slaman and Steel (Martin's conjecture, arithmetic equivalence, and countable Borel equivalence relations) that contains earlier, unpublished, as well as new results related to the theme of Part VI. Table 1 gives an overview of the papers in this volume with their original references.\n\nAs emphasized in our first two volumes, our project is not to be understood as a historical edition of old papers. In the retyping process, we uniformized and modernized notation and numbering of sections and theorems. As a consequence, references to papers in the old Cabal volumes will not always agree with references to their reprinted versions. In this volume, references to papers that already appeared in reprinted form will use the new numbering. In order to help the reader to easily cross-reference old and new numberings, we provide a list of changes after the preface.", "doi": "10.1017/CBO9781139519694.001", "isbn": "9781139519694", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2016", "pages": "vii-x" }, { "id": "authors:wwmkh-2yk50", "collection": "authors", "collection_id": "wwmkh-2yk50", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180806-095451417", "type": "book", "title": "Ordinal Definability and Recursion Theory: The Cabal Seminar", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "abstract": "The proceedings of the Los Angeles Caltech-UCLA 'Cabal Seminar' were originally published in the 1970s and 1980s. Ordinal Definability and Recursion Theory is the third in a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics and discussion of research developments since the publication of the original volumes. Focusing on the subjects of 'HOD and its Local Versions' (Part V) and 'Recursion Theory' (Part VI), each of the two sections is preceded by an introductory survey putting the papers into present context. These four volumes will be a necessary part of the book collection of every set theorist.", "doi": "10.1017/CBO9781139519694", "isbn": "9781139519694", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2016" }, { "id": "authors:jeks1-rav12", "collection": "authors", "collection_id": "jeks1-rav12", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20150107-075104911", "type": "article", "title": "Random orderings and unique ergodicity of automorphism groups", "author": [ { "family_name": "Angel", "given_name": "Omer", "clpid": "Angel-O" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Lyons", "given_name": "Russell", "clpid": "Lyons-R" } ], "abstract": "We show that the only random orderings of finite graphs that are invariant under isomorphism and induced subgraph are the uniform random orderings. We show how this implies the unique ergodicity of the automorphism group of the random graph. We give similar theorems for other structures, including, for example, metric spaces. These give the first examples of uniquely ergodic groups, other than compact groups and extremely amenable groups, after Glasner and Weiss's example of the group of all permutations of the integers. We also contrast these results to those for certain special classes of graphs and metric spaces in which such random orderings can be found that are not uniform.", "doi": "10.4171/JEMS/483", "issn": "1435-9855", "publisher": "European Mathematical Society", "publication": "Journal of the European Mathematical Society", "publication_date": "2014", "series_number": "10", "volume": "16", "issue": "10", "pages": "2059-2095" }, { "id": "authors:fsang-0q918", "collection": "authors", "collection_id": "fsang-0q918", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20131216-115036936", "type": "article", "title": "Stationary probability measures and topological realizations", "author": [ { "family_name": "Conley", "given_name": "Clinton T.", "clpid": "Conley-C-T" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Miller", "given_name": "Benjamin D.", "clpid": "Miller-B-D" } ], "abstract": "We establish the generic inexistence of stationary Borel probability measures for aperiodic Borel actions of countable groups on Polish spaces. Using this, we show that every aperiodic continuous action of a countable group on a compact Polish space has an invariant Borel set on which it has no \u03c3-compact realization.", "doi": "10.1007/s11856-013-0025-8", "issn": "0021-2172", "publisher": "Springer", "publication": "Israel Journal of Mathematics", "publication_date": "2013-11", "series_number": "1", "volume": "198", "issue": "1", "pages": "333-345" }, { "id": "authors:22xvc-qcf35", "collection": "authors", "collection_id": "22xvc-qcf35", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130405-083026269", "type": "article", "title": "Ultraproducts of measure preserving actions and graph combinatorics", "author": [ { "family_name": "Conley", "given_name": "Clinton T.", "clpid": "Conley-C-T" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Tucker-Drob", "given_name": "Robin D.", "clpid": "Tucker-Drob-R-D" } ], "abstract": "Ultraproducts of measure preserving actions of countable groups are used to study the graph combinatorics associated with such actions, including chromatic, independence and matching numbers. Applications are also given to the theory of random colorings of Cayley graphs and sofic actions and equivalence relations.", "doi": "10.1017/S0143385711001143", "issn": "0143-3857", "publisher": "Cambridge University Press", "publication": "Ergodic Theory and Dynamical Systems", "publication_date": "2013-04", "volume": "33", "pages": "334-374" }, { "id": "authors:5fpz2-tn671", "collection": "authors", "collection_id": "5fpz2-tn671", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-131021822", "type": "article", "title": "Measurable chromatic and independence numbers for ergodic\n graphs and group actions", "author": [ { "family_name": "Conley", "given_name": "Clinton T.", "clpid": "Conley-C-T" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We study in this paper combinatorial problems concerning graphs generated by measure preserving actions of countable groups on standard measure spaces. In particular we study chromatic and independence numbers, in both the measure-theoretic and the Borel context, and relate the behavior of these parameters to properties of the acting group such as amenability, Kazhdan's property (T), and freeness. We also prove a Borel analog of the classical Brooks' Theorem in finite combinatorics for actions of groups with finitely many ends.", "doi": "10.4171/GGD/179", "issn": "1661-7207", "publisher": "European Mathematical Society", "publication": "Groups, Geometry, and Dynamics", "publication_date": "2013", "series_number": "1", "volume": "7", "issue": "1", "pages": "127-180" }, { "id": "authors:d2wb1-23129", "collection": "authors", "collection_id": "d2wb1-23129", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180803-160520242", "type": "book_section", "title": "Dynamics of non-archimedean Polish groups", "book_title": "European Congress of Mathematics, Krak\u00f3w, 2-7 July, 2012", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Lata\u0142a", "given_name": "Rafa\u0142", "clpid": "Lata\u0142a-R" }, { "family_name": "Ruci\u0144ski", "given_name": "Andrzej", "clpid": "Ruci\u0144ski-A" }, { "family_name": "Strzelecki", "given_name": "Pawe\u0142", "clpid": "Strzelecki-P" }, { "family_name": "\u015awi\u0105tkowski", "given_name": "Jacek", "clpid": "\u015awi\u0105tkowski-J" }, { "family_name": "Wrzosek", "given_name": "Dariusz", "clpid": "Wrzosek-D" }, { "family_name": "Zakrzewski", "given_name": "Piotr", "clpid": "Zakrzewski-P" } ], "abstract": "A topological group G is Polish if its topology admits a compatible separable complete metric. Such a group is non-archimedean if it has a basis at the identity that consists of open subgroups. This class of Polish groups includes the profinite groups and (\u211a_p, +)\nbut our main interest here will be on non-locally compact groups. \n\nIn recent years there has been considerable activity in the study of the dynamics of Polish non-archimedean groups and this has led to interesting interactions between logic, finite combinatorics, group theory, topological dynamics, ergodic theory and representation theory. In this paper I will give a survey of some of the main directions in this area of research.", "doi": "10.4171/120", "isbn": "978-3-03719-120-0", "publisher": "European Mathematical Society", "place_of_publication": "Zurich, Switzerland", "publication_date": "2012-07", "pages": "375-397" }, { "id": "authors:0k1qp-a0g45", "collection": "authors", "collection_id": "0k1qp-a0g45", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20120716-102713726", "type": "article", "title": "Weak containment in the space of actions of a free group", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "It is shown that the translation action of the free group with n generators\non its profinite completion is the maximum, in the sense of weak containment,\nmeasure preserving action of this group. Using also a result of\nAb\u00e9rt\u2013Nikolov this is used to give a new proof of Gaboriau's theorem that\nthe cost of this group is equal to n. A similar maximality result is proved\nfor generalized shift actions. Finally a study is initiated of the class of\nresidually finite, countable groups for which the finite actions are dense in\nthe space of measure preserving actions.", "doi": "10.1007/s11856-011-0182-6", "issn": "0021-2172", "publisher": "Springer", "publication": "Israel Journal of Mathematics", "publication_date": "2012-06", "series_number": "1", "volume": "189", "issue": "1", "pages": "461-507" }, { "id": "authors:nbhb2-rc397", "collection": "authors", "collection_id": "nbhb2-rc397", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130301-104948205", "type": "article", "title": "Trigonometric series and set theory", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "This is a short historical survey concerning the interactions between the\ntheory of trigonometric series and descriptive set theory. We concentrate\nhere on the area related to problems of uniqueness for trigonometric\nseries. Detailed historical and bibliographical references can be found in the\nbooks and survey papers listed at the end.", "issn": "2080-5519", "publisher": "Polskie Towarzystwo Matematyczne", "publication": "Roczniki. Seria 2: Wiadomo\u015bci Matematyczne", "publication_date": "2012", "series_number": "2", "volume": "48", "issue": "2", "pages": "109-118" }, { "id": "authors:dqgqx-nvk43", "collection": "authors", "collection_id": "dqgqx-nvk43", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20121126-103335875", "type": "article", "title": "Dynamical properties of the automorphism groups of the random poset and random distributive lattice", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Soki\u0107", "given_name": "Miodrag", "clpid": "Soki\u0107-M" } ], "abstract": "A method is developed for proving non-amenability of certain automorphism groups of countable structures and is used to show that the automorphism groups of the random poset and random distributive lattice are not amenable. The universal minimal flow of the automorphism group of the random distributive lattice is computed as a canonical space of linear orderings but it is also shown that the class of finite distributive lattices does not admit hereditary order expansions with the Amalgamation Property.", "doi": "10.4064/fm218-1-4", "issn": "0016-2736", "publisher": "Institute of Mathematics, Polish Academy of Sciences", "publication": "Fundamenta Mathematicae", "publication_date": "2012", "series_number": "1", "volume": "218", "issue": "1", "pages": "69-94" }, { "id": "authors:8rxb0-07b70", "collection": "authors", "collection_id": "8rxb0-07b70", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-151353411", "type": "book_section", "title": "The complexity of classification problems in ergodic theory", "book_title": "Appalachian Set Theory: 2006-2012", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Tucker-Drob", "given_name": "Robin D.", "clpid": "Tucker-Drob-R-D" } ], "contributor": [ { "family_name": "Cummings", "given_name": "James", "clpid": "Cummings-J" }, { "family_name": "Schimmerling", "given_name": "Ernest", "clpid": "Schimmerling-E" } ], "abstract": "The twelfth Appalachian Set Theory workshop was held at Vanderbilt University in Nashville on October 30, 2010. The lecturer was Alexander S. Kechris. As a graduate student Robin D. Tucker-Drob assisted in writing this chapter, which is based on the workshop lectures. \n\nDedicated to the memory of Greg Hjorth (1963\u20132011) \n\nThe last two decades have seen the emergence of a theory of set theoretic complexity of classification problems in mathematics. In these lectures we will discuss recent developments concerning the application of this theory to classification problems in ergodic theory. \n\nThe first lecture will be devoted to a general introduction to this area. The next two lectures will give the basics of Hjorth's theory of turbulence, a mixture of topological dynamics and descriptive set theory, which is a basic tool for proving strong non-classification theorems in various areas of mathematics. \n\nIn the last three lectures, we will show how these ideas can be applied in proving a strong non-classification theorem for orbit equivalence. Given a countable group \u0393, two free, measure-preserving, ergodic actions of \u0393 on standard probability spaces are called orbit equivalent if, roughly speaking, they have the same orbit spaces. More precisely this means that there is an isomorphism of the underlying measure spaces that takes the orbits of one action to the orbits of the other. A remarkable result of Dye and Ornstein-Weiss asserts that any two such actions of amenable groups are orbit equivalent.", "doi": "10.1017/CBO9781139208574.010", "isbn": "9781139843065", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2012", "pages": "265-300" }, { "id": "authors:s7r86-ceg31", "collection": "authors", "collection_id": "s7r86-ceg31", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104143981", "type": "book_section", "title": "Preface", "book_title": "Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Schmid", "given_name": "Benedikt", "clpid": "Schmid-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "abstract": "This book continues the series of volumes containing reprints of the papers in the original Cabal Seminar volumes of the Springer Lecture Notes in Mathematics series [Cabal i, Cabal ii, Cabal iii, Cabal iv], unpublished material, and new papers. The first volume, [Cabal I], contained papers on games, scales and Suslin cardinals. In this volume, we continue with Parts III and IV of the project: Wadge degrees and pointclasses and Projective ordinals. As in our first volume, each of the parts contains an introductory survey (written by Alessandro Andretta and Alain Louveau for Part III and by Steve Jackson for Part IV) putting the papers into a present-day context. \n\nIn addition to the reprinted papers, this volume contains papers by Steel (More measures from AD) and Martin (Projective sets and cardinal numbers) that date back to the period of the original Cabal publications but were not included in the old volumes. Jackson contributed a new paper Regular cardinals without the weak partition property with recent results that fit well with the topic of Part IV. The paper Early investigations of the degrees of Borel sets by Wadge is a historical overview of the process of the development of the basic theory of the Wadge degrees. Table 1 gives an overview of the papers in this volume with their original references. \n\nAs emphasized in our first volume, our project is not to be understood as a historical edition of old papers. In the retyping process, we uniformized and modernized notation and numbering of sections and theorems.", "doi": "10.1017/cbo9781139028073.001", "isbn": "9781139028073", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2011-12-14", "pages": "ix-xii" }, { "id": "authors:pm2th-6kq30", "collection": "authors", "collection_id": "pm2th-6kq30", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104145629", "type": "book", "title": "Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II", "book_title": "Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Steel", "given_name": "John R." }, { "family_name": "L\u00f6we", "given_name": "Benedikt" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "abstract": "The proceedings of the Los Angeles Caltech-UCLA 'Cabal Seminar' were originally published in the 1970s and 1980s. Wadge Degrees and Projective Ordinals is the second of a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics and discussion of research developments since the publication of the original volumes. Focusing on the subjects of 'Wadge Degrees and Pointclasses' (Part III) and 'Projective Ordinals' (Part IV), each of the two sections is preceded by an introductory survey putting the papers into present context. These four volumes will be a necessary part of the book collection of every set theorist.", "doi": "10.1017/cbo9781139028073", "isbn": "9781139028073", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2011-12-14" }, { "id": "authors:8k8fg-wre98", "collection": "authors", "collection_id": "8k8fg-wre98", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104143314", "type": "book_section", "title": "A note on Wadge degrees", "book_title": "Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9781139028073.004", "isbn": "9781139028073", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2011-12-14", "pages": "43-46" }, { "id": "authors:8mahr-21p13", "collection": "authors", "collection_id": "8mahr-21p13", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180816-160145289", "type": "book_section", "title": "Homogeneous trees and projective scales", "book_title": "Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "doi": "10.1017/cbo9781139028073.014", "isbn": "9781139028073", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2011-12-14", "pages": "270-303" }, { "id": "authors:ywh08-x3k15", "collection": "authors", "collection_id": "ywh08-x3k15", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104143730", "type": "book_section", "title": "Original Numbering", "book_title": "Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9781139028073.021", "isbn": "9781139028073", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2011-12-14", "pages": "xiii-xxii" }, { "id": "authors:3rb7n-cbz70", "collection": "authors", "collection_id": "3rb7n-cbz70", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180816-103208698", "type": "article", "title": "In Memoriam: Gregory Hjorth, 1963-2011", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "Greg Hjorth suddenly and unexpectedly passed away on January 13, 2011 in Melbourne, at the age of 47, due to a heart attack. He was a remarkable person, a chess prodigy who competed internationally at a high level until his early 20s, then devoted himself to the study of philosophy and mathematics and went on to become a leading figure in the field of mathematical logic and its applications.", "issn": "1079-8986", "publisher": "Association for Symbolic Logic", "publication": "Bulletin of Symbolic Logic", "publication_date": "2011-09", "series_number": "3", "volume": "17", "issue": "3", "pages": "471-477" }, { "id": "authors:03hke-k0r77", "collection": "authors", "collection_id": "03hke-k0r77", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-140140332", "type": "book_section", "title": "The axiom of determinacy and the prewellordering property", "book_title": "Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Solovay", "given_name": "Robert M.", "clpid": "Solovay-R-M" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "abstract": "[no abstract]", "doi": "10.1017/CBO9781139028073.008", "isbn": "9780521762038", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2011", "pages": "118-140" }, { "id": "authors:eyg3y-28z07", "collection": "authors", "collection_id": "eyg3y-28z07", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-084248820", "type": "book_section", "title": "AD and projective ordinals", "book_title": "Wadge Degrees and Projective Ordinals: The Cabal Seminar, Volume II", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "abstract": "[no abstract]", "doi": "10.1017/CBO9781139028073.015", "isbn": "9780521762038", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2011", "pages": "304-345" }, { "id": "authors:49b17-8qt63", "collection": "authors", "collection_id": "49b17-8qt63", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20090908-083705826", "type": "article", "title": "Subequivalence relations and positive-definite functions", "author": [ { "family_name": "Ioana", "given_name": "Adrian", "clpid": "Ioana-A" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Tsankov", "given_name": "Todor", "clpid": "Tsankov-T" } ], "abstract": "We study a positive-definite function associated with a countable, measure-preserving equivalence relation, which can be used to measure quantitatively the proximity of subequivalence relations. Combined with a co-inducing construction introduced by Epstein and earlier work of Ioana, this can be used to construct many mixing actions of countable groups and establish the non-classifiability, in a strong sense, of orbit equivalence of actions of non-amenable groups. We also discuss connections with percolation on Cayley graphs and the theory of costs.", "doi": "10.48550/arXiv.0806.0430", "issn": "1661-7207", "publisher": "European Mathematical Society", "publication": "Groups, Geometry, and Dynamics", "publication_date": "2009", "series_number": "4", "volume": "3", "issue": "4", "pages": "579-625" }, { "id": "authors:8kgpe-kdr68", "collection": "authors", "collection_id": "8kgpe-kdr68", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130731-151623397", "type": "book_section", "title": "The equivalence of partition properties and determinacy", "book_title": "The Cabal Seminar. Volume I - Games, Scales, and Suslin Cardinals", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Woodin", "given_name": "W. Hugh", "clpid": "Woodin-W-H" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "abstract": "This paper was circulated in handwritten form in March\n1982 and contained Sections 1-4 below. There is an additional Section 5\ncontaining information about the solution of a problem mentioned in the last\nparagraph of Section I.", "isbn": "9780521899512", "publisher": "Association for Symbolic Logic", "place_of_publication": "Chicago, IL", "publication_date": "2008-09", "pages": "355-378" }, { "id": "authors:dbhn2-1d867", "collection": "authors", "collection_id": "dbhn2-1d867", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130731-150603194", "type": "book_section", "title": "Generic codes for uncountable ordinals, partition properties, and elementary embeddings", "book_title": "The Cabal Seminar. Volume I - Games, Scales, and Suslin Cardinals", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Woodin", "given_name": "W. Hugh", "clpid": "Woodin-W-H" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "abstract": "This paper was circulated in handwritten form in December 1980 and contained Sections 1-7 below. There are two additionalSections 8 and 9 here that contain further material and comments.", "isbn": "9780521899512", "publisher": "Association for Symbolic Logic", "place_of_publication": "Chicago, IL", "publication_date": "2008-09", "pages": "379-397" }, { "id": "authors:88p4a-x6g86", "collection": "authors", "collection_id": "88p4a-x6g86", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130517-111643755", "type": "article", "title": "Means on equivalence relations", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Miller", "given_name": "Benjamin D.", "clpid": "Miller-B-D" } ], "abstract": "Suppose that X is a Polish space and E is a countable Borel equivalence relation on X. We show that if there is a Borel assignment of means to the equivalence classes of E, then E is smooth. We also show that if there is a Baire measurable assignment of means to the equivalence classes of E, then E is generically smooth.", "doi": "10.1007/s11856-008-0011-8", "issn": "0021-2172", "publisher": "Springer Verlag", "publication": "Israel Journal of Mathematics", "publication_date": "2008-01", "series_number": "1", "volume": "163", "issue": "1", "pages": "241-262" }, { "id": "authors:fer92-yx674", "collection": "authors", "collection_id": "fer92-yx674", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104144489", "type": "book_section", "title": "Suslin cardinals, k-Suslin sets, and the scale property in the hyperprojective hierarchy", "book_title": "Games, Scales and Suslin Cardinals: The Cabal Seminar, Volume I", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9780511546488.016", "isbn": "9780511546488", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2008", "pages": "314-332" }, { "id": "authors:2zkan-3mb89", "collection": "authors", "collection_id": "2zkan-3mb89", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:KECpams08", "type": "article", "title": "Amenable actions and almost invariant sets", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Tsankov", "given_name": "Todor", "clpid": "Tsankov-T" } ], "abstract": "In this paper, we study the connections between properties of the action of a countable group \u0393 on a countable set X and the ergodic theoretic properties of the corresponding generalized Bernoulli shift, i.e., the corresponding shift action of \u0393 on M^X, where M is a measure space. In particular, we show that the action of \u0393 on X is amenable iff the shift \u0393 \u2192 M^X has almost invariant sets.", "doi": "10.1090/S0002-9939-07-09116-2", "issn": "0002-9939", "publisher": "American Mathematical Society", "publication": "Proceedings of the American Mathematical Society", "publication_date": "2008", "series_number": "2", "volume": "136", "issue": "2", "pages": "687-697" }, { "id": "authors:rj56p-qfv15", "collection": "authors", "collection_id": "rj56p-qfv15", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-133923114", "type": "book_section", "title": "A coding theorem for measures", "book_title": "Games, Scales and Suslin Cardinals: The Cabal Seminar, Volume I", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "abstract": "[no abstract]", "doi": "10.1017/CBO9780511546488.020", "isbn": "9780521899512", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2008", "pages": "398-403" }, { "id": "authors:ckzr1-hh534", "collection": "authors", "collection_id": "ckzr1-hh534", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-080825868", "type": "book", "title": "Games, Scales and Suslin Cardinals: The Cabal Seminar, Volume I", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt" }, { "family_name": "Steel", "given_name": "John R." } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "abstract": "The proceedings of the Los Angeles Caltech-UCLA 'Cabal Seminar' were originally published in the 1970s and 1980s. Games, Scales, and Suslin Cardinals is the first of a series of four books collecting the seminal papers from the original volumes together with extensive unpublished material, new papers on related topics, and discussion of research developments since the publication of the original volumes. Focusing on the subjects of 'Games and Scales' (Part 1) and 'Suslin Cardinals, Partition Properties, and Homogeneity' (Part 2), each of the two sections is preceded by an introductory survey putting the papers into present context. This volume will be an invaluable reference for anyone interested in higher set theory.", "doi": "10.1017/CBO9780511546488", "isbn": "9780521899512", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2008" }, { "id": "authors:4zqa3-gb520", "collection": "authors", "collection_id": "4zqa3-gb520", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104143442", "type": "book_section", "title": "Notes on the theory of scales", "book_title": "Games, Scales and Suslin Cardinals: The Cabal Seminar, Volume I", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Moschovakis", "given_name": "Yiannis N.", "clpid": "Moschovakis-Y-N" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9780511546488.003", "isbn": "9780511546488", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2008", "pages": "28-74" }, { "id": "authors:6h1q9-t1a76", "collection": "authors", "collection_id": "6h1q9-t1a76", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-141457603", "type": "book_section", "title": "The axiom of determinacy, strong partition properties, and nonsingular measures", "book_title": "Games, Scales and Suslin Cardinals: The Cabal Seminar, Volume I", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Kleinber", "given_name": "Eugene M.", "clpid": "Kleinber-E-M" }, { "family_name": "Moschovakis", "given_name": "Yiannis N.", "clpid": "Moschovakis-Y-N" }, { "family_name": "Woodin", "given_name": "W. Hugh", "clpid": "Woodin-W-H" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "abstract": "[no abstract]", "doi": "10.1017/CBO9780511546488.017", "isbn": "9780521899512", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2008", "pages": "333-354" }, { "id": "authors:58ecr-dha68", "collection": "authors", "collection_id": "58ecr-dha68", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104144069", "type": "book_section", "title": "Preface", "book_title": "Games, Scales and Suslin Cardinals: The Cabal Seminar, Volume I", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "L\u00f6we", "given_name": "Benedikt", "clpid": "L\u00f6we-B" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "abstract": "This is the first of four volumes containing reprints of the papers in the original Cabal Seminar volumes of the Springer Lecture Notes in Mathematics series [cabal i, cabal ii, cabal iii, cabal iv], unpublished material, and new papers. We have grouped the papers of the original Cabal Seminar volumes according to their topics. This volume contains the papers on \"Games and Scales\" (Part I) and \"Suslin Cardinals, Partition Properties, Homogeneity\" (Part II). Each of the parts contains an introductory survey (written by John Steel and Steve Jackson, respectively) putting the papers into a present-day context. Table 1 gives an overview of the papers in this volume with their original references. \n\nThis volume must not be understood as a historical edition of old papers. In the 1980s, there were a number of results obtained by the researchers associated with the Cabal Seminar, some of which were intended for a fifth Cabal volume that was never published. We include some of these papers in this volume, together with papers reporting on new developments related to the research of the Cabal Seminar. These papers are Steel's \"Scales in K(\u211d)\" and \"The Length-\u03c91 Open Game Quantifier Propagates Scales\" in Part Iand \"The Equivalence of Partition Properties and Determinacy\" and \"Generic Codes for Uncountable Ordinals, Partition Properties, and Elementary Embeddings\" by Kechris and Woodin, \"The Tree of a Moschovakis Scale is Homogeneous\" by Martin and Steel, and \"Weakly Homogeneous Trees\" by Martin and Woodin in Part II.", "doi": "10.1017/cbo9780511546488.001", "isbn": "9780511546488", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2008", "pages": "ix-xi" }, { "id": "authors:34s0v-by239", "collection": "authors", "collection_id": "34s0v-by239", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-084128445", "type": "article", "title": "Turbulence, amalgamation, and generic automorphisms of homogeneous structures", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Rosendal", "given_name": "Christian", "clpid": "Rosendal-C" } ], "abstract": "We study topological properties of conjugacy classes in Polish groups, with emphasis on automorphism groups\nof homogeneous countable structures. We first consider the existence of dense conjugacy classes (the topological\nRokhlin property). We then characterize when an automorphism group admits a comeager conjugacy class\n(answering a question of Truss) and apply this to show that the homeomorphism group of the Cantor space has\na comeager conjugacy class (answering a question of Akin, Hurley and Kennedy). Finally, we study Polish groups\nthat admit comeager conjugacy classes in any dimension (in which case the groups are said to admit ample\ngenerics). We show that Polish groups with ample generics have the small index property (generalizing results\nof Hodges, Hodkinson, Lascar and Shelah) and arbitrary homomorphisms from such groups into separable\ngroups are automatically continuous. Moreover, in the case of oligomorphic permutation groups, they have\nuncountable cofinality and the Bergman property. These results in particular apply to automorphism groups of\nmany \u03c9-stable, \u21350-categorical structures and of the random graph. In this connection, we also show that the\ninfinite symmetric group S\u221e has a unique non-trivial separable group topology. For several interesting groups\nwe also establish Serre's properties (FH) and (FA).", "doi": "10.1112/plms/pdl007", "issn": "0024-6115", "publisher": "London Mathematical Society", "publication": "Proceedings of the London Mathematical Society", "publication_date": "2007-03-01", "series_number": "2", "volume": "94", "issue": "2", "pages": "302-350" }, { "id": "authors:wwqtw-d9y58", "collection": "authors", "collection_id": "wwqtw-d9y58", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130612-132700025", "type": "article", "title": "Unitary representations and modular actions", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We call a measure-preserving action of a countable discrete group on a standard probability space tempered if the associated Koopman representation restricted to the orthogonal complement to the constant functions is weakly contained in the regular representation. Extending a result of Hjorth, we show that every tempered action is antimodular, i.e., in a precise sense \"orthogonal\" to any Borel action of a countable group by automorphisms on a countable rooted tree. We also study tempered actions of countable groups by automorphisms on compact metrizable groups, where it turns out that this notion has several ergodic theoretic reformulations and fits naturally in a hierarchy of strong ergodicity properties strictly between ergodicity and strong mixing.", "doi": "10.1007/s10958-007-0449-y", "issn": "1072-3374", "publisher": "Springer Verlag", "publication": "Journal of Mathematical Sciences", "publication_date": "2007-01", "series_number": "3", "volume": "140", "issue": "3", "pages": "398-425" }, { "id": "authors:0xfe3-hcw27", "collection": "authors", "collection_id": "0xfe3-hcw27", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130610-145818965", "type": "article", "title": "Rigidity theorems for actions of product groups and countable Borel equivalence relations", "author": [ { "family_name": "Hjorth", "given_name": "Greg", "clpid": "Hjorth-G" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "This Memoir is both a contribution to the theory of Borel equivalence relations,\nconsidered up to Borel reducibility, and measure preserving group actions\nconsidered up to orbit equivalence. Here E is said to be Borel reducible to F if\nthere is a Borel function f with xEy if and only if f(x)Ff(y). Moreover, E is orbit\nequivalent to F if the respective measure spaces equipped with the extra structure\nprovided by the equivalence relations are almost everywhere isomorphic.\nWe consider product groups acting ergodically and by measure preserving transformations\non standard Borel probability spaces. In general terms, the basic parts\nof the monograph show that if the groups involved have a suitable notion of \"boundary\" (we make this precise with the definition of near hyperbolic), then one orbit\nequivalence relation can only be Borel reduced to another if there is some kind of\nalgebraic resemblance between the product groups and coupling of the action. This\nalso has consequence for orbit equivalence. In the case that the original equivalence\nrelations do not have non-trivial almost invariant sets, the techniques lead to relative\nergodicity results. An equivalence relation E is said to be relatively ergodic to\nF if any f with xEy\u21d2 f(x)Ff(y) has [f(x)]F constant almost everywhere.\nThis underlying collection of lemmas and structural theorems is employed in a\nnumber of different ways.\nOne of the most pressing concerns was to give completely self-contained proofs\nof results which had previously only been obtained using Zimmer's superrigidity\ntheory. We present \"elementary proofs\" that there are incomparable countable\nBorel equivalence relations (Adams-Kechris), inclusion does not imply reducibility\n(Adams), and (n + 1)E is not necessarily reducible to nE (Thomas).\nIn the later parts of the paper we give applications of the theory to specific\ncases of product groups. In particular, we catalog the actions of products of the\nfree group and obtain additional rigidity theorems and relative ergodicity results in\nthis context.\nThere is a rather long series of appendices, whose primary goal is to give the\nreader a comprehensive account of the basic techniques. But included here are also\nsome new results. For instance, we show that the Furstenberg-Zimmer lemma on\ncocycles from amenable groups fails with respect to Baire category, and use this\nto answer a question of Weiss. We also present a different proof that F_2 has the\nHaagerup approximation property.", "issn": "0065-9266", "publisher": "American Mathematical Society", "publication": "Memoirs of the American Mathematical Society", "publication_date": "2005-09", "series_number": "833", "volume": "177", "issue": "833", "pages": "1-109" }, { "id": "authors:9r886-m4x65", "collection": "authors", "collection_id": "9r886-m4x65", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130517-135759630", "type": "article", "title": "Fra\u00efss\u00e9 Limits, Ramsey Theory, and topological dynamics of automorphism groups", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Pestov", "given_name": "V. G.", "clpid": "Pestov-P-G" }, { "family_name": "Todorcevic", "given_name": "S.", "clpid": "Todorcevic-S" } ], "abstract": "(A) In this paper we study some connections between the Fra\u00efss\u00e9 theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures. A prime concern of topological dynamics is the study of continuous actions of (Hausdorff) topological groups G on (Hausdorff) compact spaces X.", "doi": "10.1007/s00039-005-0503-1", "issn": "1016-443X", "publisher": "Springer", "publication": "Geometric and Functional Analysis", "publication_date": "2005-02-01", "series_number": "1", "volume": "15", "issue": "1", "pages": "106-189" }, { "id": "authors:4fg28-3pc95", "collection": "authors", "collection_id": "4fg28-3pc95", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130612-095807587", "type": "book_section", "title": "Actions of Polish Groups and Classification Problems", "book_title": "Analysis and Logic", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Henson", "given_name": "Ward C.", "clpid": "Henson-W-C" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" }, { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "We will discuss in this paper some aspects of a general program whose goal is the development of the theory of definable actions of Polish groups, the structure and classification of their orbit spaces, and the closely related study of definable equivalence relations. This work is motivated by basic foundational questions, like understanding the nature of complete classification of mathematical objects up to some notion of equivalence by invariants, and creating a mathematical framework for\nmeasuring the complexity of such classification problems. This theory, which has been growing rapidly over the last few years, is developed within the context of descriptive set theory, which provides the basic underlying concepts and methods. On the other hand, in view of the broad scope of this theory, there are natural interactions of it with other areas of mathematics, such as the theory of topological groups, topological dynamics, ergodic theory and its relationships with the theory of operator algebras, model theory, and recursion theory.", "isbn": "9780521648615", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003-01", "pages": "115-187" }, { "id": "authors:cgh0x-enw90", "collection": "authors", "collection_id": "cgh0x-enw90", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130516-152757565", "type": "article", "title": "On the classification of Polish metric spaces up to isometry", "author": [ { "family_name": "Gao", "given_name": "Su", "clpid": "Gao-Su" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We study the classification problem of Polish metric spaces up to isometry and\nthe isometry groups of Polish metric spaces. In the framework of the descriptive\nset theory of definable equivalence relations, we determine the exact complexity of\nvarious classification problems concerning Polish metric spaces. We start with the\nclass of all Polish metric spaces and prove that it is Borel bireducible to the universal\norbit equivalence relation induced by Borel actions of Polish groups. We then turn\nto special classes of Polish metric spaces, including locally compact, ultrametric,\nzero-dimensional, homogeneous, and ultrahomogeneous spaces. In the investigation\nof the classification problems we also obtain characterizations for isometry groups\nof various classes of Polish metric spaces.", "issn": "0065-9266", "publisher": "American Mathematical Society", "publication": "Memoirs of the American Mathematical Society", "publication_date": "2003-01", "series_number": "766", "volume": "161", "issue": "766", "pages": "1-78" }, { "id": "authors:m2cg8-pbh82", "collection": "authors", "collection_id": "m2cg8-pbh82", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180816-160145363", "type": "book_section", "title": "Introduction", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "We will discuss in this paper some aspects of a general program whose goal is the development of the theory of definable actions of Polish groups, the structure and classification of their orbit spaces, and the closely related study of definable equivalence relations. This work is motivated by basic foundational questions, like understanding the nature of complete classification of mathematical objects up to some notion of equivalence by invariants, and creating a mathematical framework for measuring the complexity of such classification problems. This theory, which has been growing rapidly over the last few years, is developed within the context of descriptive set theory, which provides the basic underlying concepts and methods. On the other hand, in view of the broad scope of this theory, there are natural interactions of it with other areas of mathematics, such as the theory of topological groups, topological dynamics, ergodic theory and its relationships with the theory of operator algebras, model theory, and recursion theory. \n\nClassically, in various branches of dynamics one studies actions of the groups of integers \u2124, reals \u211d, Lie groups, or even more generally (second countable) locally compact groups. One of the goals of the theory is to expand this scope by considering the more comprehensive class of Polish groups (separable completely metrizable topological groups), which seems to be the widest class of well-behaved (for our purposes) groups and which includes practically every type of topological group we are interested in.", "doi": "10.1017/cbo9781107360006.017", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "117-119" }, { "id": "authors:6rb1r-8pd85", "collection": "authors", "collection_id": "6rb1r-8pd85", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104144743", "type": "book_section", "title": "Turbulence I: Overview", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9781107360006.023", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "137-142" }, { "id": "authors:8gtmg-6yp79", "collection": "authors", "collection_id": "8gtmg-6yp79", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104145310", "type": "book_section", "title": "Turbulence VII: The Second Main Theorem", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9781107360006.029", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "163-183" }, { "id": "authors:473k1-9z653", "collection": "authors", "collection_id": "473k1-9z653", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180816-160145605", "type": "book_section", "title": "Introduction", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "The most outstanding problems in the theory of infinite dimensional Banach spaces, those that were central to the study of the general structure of a Banach space, finally yielded their secrets in the 1990's. In this survey we shall discuss these problems and their solutions and more. For many years researchers have been aware of deep connections between both the theorems and ideas of logic and set theory and Banach space theory. We shall try to illuminate these connections as well.\n\nFor example the ideas of Ramsey theory played a key role in H. Rosenthal's magnificent l_1-theorem in 1974 [R1]. But there is also a less direct connection with the Banach space question as to whether or not separable infinite dimensional Hilbert space, l_2, is distortable. This is equivalent to the following approximate Ramsey problem. Let S_(l2) = {x \u2208 l_2: \u2225x\u2225 = 1} be the unit sphere of l_2. Finitely color the sphere by colors C_1,\u2026, C_k and let \u03b5 > 0. Does there exist an i_0 and an infinite dimensional closed linear subspace X of l_2 so that the unit sphere of X, S_X, is a subset of (C_(i0))_\u03b5 = {y \u2208 S_(l2) : \u2225y \u2212 x\u2225 < \u03b5 for some x \u2208 C_(i0)}? It suffices to let (e_i) be an orthonormal basis for l_2 and confine the search to block subspaces \u2014 those spanned by block bases of (e_i) (these terms are defined precisely below).", "doi": "10.1017/cbo9781107360006.031", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "191-193" }, { "id": "authors:5ttz2-yka90", "collection": "authors", "collection_id": "5ttz2-yka90", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-150243521", "type": "book_section", "title": "Basic Model Theory I", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/CBO9781107360006.008", "isbn": "9780521648615", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "31-33" }, { "id": "authors:9q2qp-vhe34", "collection": "authors", "collection_id": "9q2qp-vhe34", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104145215", "type": "book_section", "title": "Turbulence VI: The First Main Theorem", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9781107360006.028", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "157-162" }, { "id": "authors:9nata-d1x39", "collection": "authors", "collection_id": "9nata-d1x39", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180816-160145084", "type": "book_section", "title": "Gowers' dichotomy: A block Ramsey Theorem", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9781107360006.034", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "217-222" }, { "id": "authors:3a566-2z615", "collection": "authors", "collection_id": "3a566-2z615", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-082937079", "type": "book_section", "title": "Actions of the Infinite Symmetric Group", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/CBO9781107360006.022", "isbn": "9780521648615", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "133-136" }, { "id": "authors:a13d8-j4n92", "collection": "authors", "collection_id": "a13d8-j4n92", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-081922189", "type": "book_section", "title": "Actions of Locally Compact Groups", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/CBO9781107360006.021", "isbn": "9780521648615", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "131-132" }, { "id": "authors:ap14y-xbw77", "collection": "authors", "collection_id": "ap14y-xbw77", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180815-101343549", "type": "book_section", "title": "The General Glimm-Effros Dichotomy", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/CBO9781107360006.018", "isbn": "9780521648615", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "120-122" }, { "id": "authors:1c9tj-t2y51", "collection": "authors", "collection_id": "1c9tj-t2y51", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104144921", "type": "book_section", "title": "Turbulence III: Induced Actions", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9781107360006.025", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "146-149" }, { "id": "authors:btb5z-qfg46", "collection": "authors", "collection_id": "btb5z-qfg46", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-135054931", "type": "book_section", "title": "Asymptotic structure", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/CBO9781107360006.036", "isbn": "9780521648615", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "233-243" }, { "id": "authors:z6qr5-nee80", "collection": "authors", "collection_id": "z6qr5-nee80", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104145112", "type": "book_section", "title": "Turbulence V: Calmness", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9781107360006.027", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "152-156" }, { "id": "authors:d63ag-h8e78", "collection": "authors", "collection_id": "d63ag-h8e78", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104145409", "type": "book_section", "title": "Types", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9781107360006.016", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "96-106" }, { "id": "authors:dsej0-d5z80", "collection": "authors", "collection_id": "dsej0-d5z80", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104143625", "type": "book_section", "title": "Ordinal Indices", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9781107360006.037", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "244-251" }, { "id": "authors:yaajk-0wk55", "collection": "authors", "collection_id": "yaajk-0wk55", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180816-160145154", "type": "book_section", "title": "The homogeneous Banach space problem", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Chritian", "clpid": "Michaux-C" } ], "abstract": "We cannot end before at least briefly discussing one other spectacular result of the 90's.\n\nWe recall that the homogeneous Banach space problem (P5) is: If X is isomorphic to all Y \u2286 X, is X isomorphic to l2? This was solved by combining two beautiful pieces of work, Gowers' dichotomy theorem (Theorem 3.1) and the following theorem of Komorowski and Tomczak-Jaegermann [KT1, KT2]. A nice exposition somewhat simplifying the argument appears in [TJ1].", "doi": "10.1017/cbo9781107360006.038", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "252-253" }, { "id": "authors:e5jrc-jfm47", "collection": "authors", "collection_id": "e5jrc-jfm47", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-082427900", "type": "book_section", "title": "Actions of Polish Groups", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/CBO9781107360006.019", "isbn": "9780521648615", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "123-125" }, { "id": "authors:f9056-g4d25", "collection": "authors", "collection_id": "f9056-g4d25", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-105759569", "type": "article", "title": "Universal minimal flows of automorphism groups", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Pestov", "given_name": "V.", "clpid": "Pestov-V" }, { "family_name": "Todor\u010devi\u0107", "given_name": "S.", "clpid": "Todor\u010devi\u0107-S" } ], "abstract": "We investigate some connections between the Fra\u00efss\u00e9 theory of amalgamation classes and ultrahomogeneous structures, Ramsey theory, and topological dynamics of automorphism groups of countable structures. We show, in particular, that results from the structural Ramsey theory can be quite useful in recognizing the universal minimal flows of this kind of groups. As a result we compute the universal minimal flows of several well known topological groups such as, for example, the automorphism group of the random graph, the automorphism group of the random triangle-free graph, the automorphism group of the \u221e-dimensional vector space over a finite field, the automorphism group of the countable atomless Boolean algebra, etc. So we have here a reversal in the traditional relationship between topological dynamics and Ramsey theory: the Ramsey-theoretic results are used in proving theorems of topological dynamics rather than vice versa.", "issn": "0561-7332", "publisher": "Srpska akademija nauka i umetnosti, Beograd", "publication": "Bulletin Classe des Sciences Math\u00e9matiques et Naturelles, Sciences math\u00e9matiques", "publication_date": "2003", "volume": "28", "pages": "93-106" }, { "id": "authors:wzw19-68k90", "collection": "authors", "collection_id": "wzw19-68k90", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104145709", "type": "book_section", "title": "Ultraproducts of Normed Space Structures", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9781107360006.010", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "38-42" }, { "id": "authors:gd1r7-a4b67", "collection": "authors", "collection_id": "gd1r7-a4b67", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-085559683", "type": "book_section", "title": "Alternative Formulations of the Theory", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/CBO9781107360006.013", "isbn": "9780521648615", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "72-74" }, { "id": "authors:wdhyt-nf903", "collection": "authors", "collection_id": "wdhyt-nf903", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104145503", "type": "book_section", "title": "Ultrapowers of Normed Space Structures", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9781107360006.006", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "17-20" }, { "id": "authors:kch2y-3f752", "collection": "authors", "collection_id": "kch2y-3f752", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180816-160145233", "type": "book_section", "title": "Homogeneous Structures", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9781107360006.014", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "75-78" }, { "id": "authors:sa1wa-twg11", "collection": "authors", "collection_id": "sa1wa-twg11", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180816-160145810", "type": "book_section", "title": "More Model Theory", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9781107360006.015", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "79-95" }, { "id": "authors:m3qfe-dr555", "collection": "authors", "collection_id": "m3qfe-dr555", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-151926459", "type": "book_section", "title": "Concluding remarks", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "Summary: We have not addressed certain important problems that remain unsolved after many years concerning the classical Banach spaces themselves. (Q13) Let K be a compact metric space. Is every complemented sub-space of C(K) isomorphic to C(L) for some compact metric space L?\nIt is known that if K is uncountable then C(K) is isomorphic to C[0,1]. If if is countable then C(K) is isomorphic to C(\u03c9^(\u03c9\u03b1)) for some \u03b1 < \u03c9_1. Every complemented subspace of c_0 (isomorphic to C (\u03c9)) is either finite dimensional or isomorphic to c_0 ([Pel]). If X is complemented in C[0,1] and X* is nonseparable then X is isomorphic to C[0,1] [R6]. Every quotient of c0 embeds isomorphically into c0 but this does not hold in general for C(\u03c9^(\u03c9\u03b1)). A discussion of these and related results may be found in [A1, A2, A3, A4], [Gal, Ga2], [Bo2].\nThe isomorphism types of the complemented subspaces of L_1[0,1] remain unclassified.\n(Q14) Let X be a complemented (infinite dimensional) subspace of L_1[0,1]. Is X isomorphic to L_1 or l_1?\nEvery X which is complemented in l_p (1 \u2264 p < \u221e) or c_0 is isomorphic to l_p or c_0. There are known to be uncountably many mutually nonisomorphic complemented subspaces of L_p[0,1] (1 < p < \u221e, p \u2260 2) [BRS] and all are known to have a basis [JRZ]. These spaces have been classified as \u2112_p spaces ([LP], [LR]), provided they are not Hilbert spaces.", "doi": "10.1017/CBO9781107360006.039", "isbn": "9780521648615", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "254-256" }, { "id": "authors:m6d1f-n8p96", "collection": "authors", "collection_id": "m6d1f-n8p96", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-135928044", "type": "book_section", "title": "Actions of Countable Groups", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "Let G be a countable group and X a Borel G-space. Then it is clear that is a Borel equivalence relation and every one of its equivalence class is countable.", "doi": "10.1017/CBO9781107360006.020", "isbn": "9780521648615", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "126-130" }, { "id": "authors:rjbbe-mby02", "collection": "authors", "collection_id": "rjbbe-mby02", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104145011", "type": "book_section", "title": "Turbulence IV: Some Examples", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9781107360006.026", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "150-151" }, { "id": "authors:q9tk3-c2q65", "collection": "authors", "collection_id": "q9tk3-c2q65", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104143887", "type": "book_section", "title": "Positive Bounded Formulas", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "In this chapter we introduce the key ingredients of the logic for normed space structures that is described in this paper. These are the positive bounded formulas and the concept of approximate satisfaction of such formulas in normed space structures. \n\nLet L be a signature for a normed space structure \u2133 based on (M^(s) \u2223 s \u2208 S). Recall that S has a distinguished element s = S_\u211d for which M^(s) = \u211d is the sort of real numbers.\n\nWe begin considering \u2133 from the model theoretic point of view, introducing a formal language based on L and a semantics according to which this language is interpreted in \u2133. In addition to the symbols of the signature L, we also need for each element s of the sort index set S, a countable set of symbols called the variables of sort s. \n\nWe begin defining the formal language by introducing the set of terms of L, or L-terms. Each term is a finite string of symbols, each of which may be a variable or a function symbol of L, or one of the symbols (or, which are used for punctuation. In this many-sorted context, each term is associated with a unique sort which indicates its range. The formal definition is recursive. \n\nDefinition. An L-term with range of sort s is a string which can be obtained by finitely many applications of the following rules of formation: \n\nIf x is a variable of L of sort s, then x is a term with range of sort s. \n\n[\u2026]", "doi": "10.1017/cbo9781107360006.007", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "21-30" }, { "id": "authors:mrt0x-cz810", "collection": "authors", "collection_id": "mrt0x-cz810", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180815-100341894", "type": "book_section", "title": "Distortion", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/CBO9781107360006.035", "isbn": "9780521648615", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "223-232" }, { "id": "authors:qanf0-fg898", "collection": "authors", "collection_id": "qanf0-fg898", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-150624583", "type": "book_section", "title": "Basic Model Theory II", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/CBO9781107360006.011", "isbn": "9780521648615", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "43-54" }, { "id": "authors:q2y8w-bg926", "collection": "authors", "collection_id": "q2y8w-bg926", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104145797", "type": "book_section", "title": "The unconditional basic sequence problem and connections with distortion", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9781107360006.033", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "208-216" }, { "id": "authors:nepa8-sy522", "collection": "authors", "collection_id": "nepa8-sy522", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104144833", "type": "book_section", "title": "Turbulence II: Basic Facts", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9781107360006.024", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "143-145" }, { "id": "authors:vx1d9-tnt24", "collection": "authors", "collection_id": "vx1d9-tnt24", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180816-160145519", "type": "book_section", "title": "Introduction", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "The articles in this book had their origins in three mini-courses offered at the conference \"Analyse & Logique\" held August 25\u201329, 1997, at the University of Mons-Hainaut in M\u00f6ns, Belgium. For a long time there have been rich connections between analysis and logic; these articles bear witness that this relationship is still very active, and continues to be important for both areas. \n\nHere we briefly describe these three articles; each one has a more detailed Introduction as its first chapter.", "doi": "10.1017/cbo9781107360006.002", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "xi-xiv" }, { "id": "authors:htjz5-56n45", "collection": "authors", "collection_id": "htjz5-56n45", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180816-160145747", "type": "book_section", "title": "Isomorphic Ultrapowers", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "In this chapter we prove isomorphism theorems for ultrapowers and ultraproducts of normed space structures. These results show that there is a very tight connection between (a) properties that are preserved under the ultraproduct construction and (b) properties that are expressible using the logic for normed space structures that is described in this paper. \n\nLet L be a signature and let \u2133 and N be two normed space L-structures. If \u2133 and N have isomorphic ultrapowers, by Corollary 9.4 they must be approximately elementarily equivalent. Theorem 10.7 below gives the converse (in a strong form). Together these results show that ultrapower equivalence of \u2133 and N is the same as approximate elementary equivalence. (See the discussion of this issue in the Introduction.) \n\nTheorem 10.8 below is a similar result for ultraproducts. Among other things, it shows that the ultrafilter guaranteed by Theorem 10.7 can be chosen in a highly uniform way and that the ultrapowers in question can be taken to be highly saturated. The uniformity will be exploited in Chapter 12 to prove the existence of ultrapowers that are highly homogeneous (in addition to being highly saturated). \n\nThe results in this chapter are analogous to the Keisler-Shelah Theorem in ordinary model theory. (See [She71] and Chapter 6 in [CK90].) Moreover, our proof follows a similar line of argument, with adjustments appropriate to the handling of positive bounded formulas and their approximations.", "doi": "10.1017/cbo9781107360006.012", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "55-71" }, { "id": "authors:j2y60-rz007", "collection": "authors", "collection_id": "j2y60-rz007", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-092445662", "type": "book", "title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine" }, { "family_name": "Michaux", "given_name": "Christian" } ], "abstract": "This volume comprises articles from four outstanding researchers who work at the cusp of analysis and logic. The emphasis is on active research topics; many results are presented that have not been published before and open problems are formulated. Considerable effort has been made by the authors to integrate their articles and make them accessible to mathematicians new to the area.", "doi": "10.1017/CBO9781107360006", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003" }, { "id": "authors:vj27x-0pw15", "collection": "authors", "collection_id": "vj27x-0pw15", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104144394", "type": "book_section", "title": "Signatures", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9781107360006.005", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "13-16" }, { "id": "authors:trwbz-0ta26", "collection": "authors", "collection_id": "trwbz-0ta26", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180816-160145891", "type": "book_section", "title": "Normed Space Structures", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9781107360006.004", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "10-12" }, { "id": "authors:t4ch5-78634", "collection": "authors", "collection_id": "t4ch5-78634", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-103955912", "type": "book_section", "title": "Background material: The 60's and 70's", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "[no abstract]", "doi": "10.1017/CBO9781107360006.032", "isbn": "9780521648615", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "194-207" }, { "id": "authors:st8kc-pcq29", "collection": "authors", "collection_id": "st8kc-pcq29", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180816-160145668", "type": "book_section", "title": "Introduction", "book_title": "Analysis and Logic", "author": [ { "family_name": "Henson", "given_name": "C. Ward", "clpid": "Henson-C-W" }, { "family_name": "Iovino", "given_name": "Jos\u00e9", "clpid": "Iovino-J" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Odell", "given_name": "Edward", "clpid": "Odell-E" } ], "contributor": [ { "family_name": "Finet", "given_name": "Catherine", "clpid": "Finet-C" }, { "family_name": "Michaux", "given_name": "Christian", "clpid": "Michaux-C" } ], "abstract": "The ideas and methods of model theory are being applied today in nearly all parts of mathematics. Here we concentrate on a framework for applications in functional analysis. Model theory has already provided several tools for the research analyst, of which the most important are: (a) the Banach space ultraproduct and nonstandard hull constructions in functional analysis; (b) spaces of (model-theoretic) types as used in the geometry of Banach spaces; and (c) Loeb measure spaces in stochastic analysis and its applications. In this paper we explain a systematic model theoretic framework within which these tools (especially (a) and (b)) are naturally situated. Our main intended audience consists of analysts who are familiar with the ultraproduct construction, and this perspective has strongly influenced our presentation of the material. We expect that many model theorists will also find something of interest in this subject. In particular, we indicate the initial steps of a program for introducing the key ideas and methods of model theory into functional analysis in a systematic and comprehensive way. \n\nApplications of model theory in functional analysis have been pursued since the mid 1960s, beginning with the introduction of Banach space ultraproducts by Bretagnolle, Dacunha-Castelle, and Krivine [BDCK66] [Kri67] [DCK72] and nonstandard hulls by Luxemburg [Lux69b]. These two constructions Avere first used at about the same time and they are essentially the same; however, these initial steps led to largely independent lines of research that have still not been fully integrated.", "doi": "10.1017/cbo9781107360006.003", "isbn": "9781107360006", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "2003", "pages": "3-9" }, { "id": "authors:6sy4f-ajc11", "collection": "authors", "collection_id": "6sy4f-ajc11", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130520-104729185", "type": "article", "title": "Countable Borel equivalence relations", "author": [ { "family_name": "Jackson", "given_name": "S.", "clpid": "Jackson-S" }, { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "A.", "clpid": "Louveau-A" } ], "abstract": "This paper develops the foundations of the descriptive set theory of countable Borel\nequivalence relations on Polish spaces with particular emphasis on the study of hyper-finite, amenable, treeable and universal equivalence relations.", "issn": "0219-0613", "publisher": "World Scientific Publishing", "publication": "Journal of Mathematical Logic", "publication_date": "2002", "series_number": "1", "volume": "2", "issue": "1", "pages": "1-80" }, { "id": "authors:egdzc-88960", "collection": "authors", "collection_id": "egdzc-88960", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111104-093324189", "type": "article", "title": "A strong generic ergodicity property of unitary and self-adjoint operators", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Sofronidis", "given_name": "N. E.", "clpid": "Sofronidis-N-E" } ], "abstract": "Consider the conjugacy action of the unitary group of an infinite-dimensional separable Hilbert space on the unitary operators. A strong generic ergodicity property of this action is established, by showing that any conjugacy invariants assigned in a definable way to unitary operators, and taking as values countable structures up to isomorphism, generically trivialize. Similar results are proved for conjugacy of self-adjoint operators and for measure equivalence. The proofs make use of the theory of turbulence for continuous actions of Polish groups, developed by Hjorth. These methods are also used to give a new solution to a problem of Mauldin in measure theory, by showing that any analytic set of pairwise orthogonal measures on the Cantor space is orthogonal to a product measure.", "doi": "10.1017/S0143385701001705", "issn": "0143-3857", "publisher": "Cambridge University Press", "publication": "Ergodic Theory and Dynamical Systems", "publication_date": "2001-10", "volume": "21", "pages": "1459-1479" }, { "id": "authors:3mgdb-g1w72", "collection": "authors", "collection_id": "3mgdb-g1w72", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130517-132314741", "type": "article", "title": "Polish Metric Spaces: Their Classification and Isometry Groups", "author": [ { "family_name": "Clemens", "given_name": "John D.", "clpid": "Clemens-J-D" }, { "family_name": "Gao", "given_name": "Su", "clpid": "Gao-Su" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "In this communication we present some recent results on the classification of Polish metric spaces up to isometry and on the isometry groups of Polish metric spaces. A Polish metric space is a complete separable metric space (X,d). Our first goal is to determine the exact complexity of the classification problem of general Polish metric spaces up to isometry. This work was motivated by a paper of Vershik [1998], where he remarks (in the beginning of Section 2): \"The classification of Polish spaces up to isometry is an enormous task. More precisely, this classification is not 'smooth' in the modern terminology.\" Our Theorem 2.1 below quantifies precisely the enormity of this task. After doing this, we turn to special classes of Polish metric spaces and investigate the classification problems associated with them. Note that these classification problems are in principle no more complicated than the general one above. However, the determination of their exact complexity is not necessarily easier. The investigation of the classification problems naturally leads to some interesting results on the groups of isometries of Polish metric spaces. We shall also present these results below.", "doi": "10.2307/2687754", "issn": "1079-8986", "publisher": "Association for Symbolic Logic", "publication": "Bulletin of Symbolic Logic", "publication_date": "2001-09", "series_number": "3", "volume": "7", "issue": "3", "pages": "361-375" }, { "id": "authors:zn712-e6r41", "collection": "authors", "collection_id": "zn712-e6r41", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-132240239", "type": "article", "title": "The prospects for mathematical logic in the twenty-first century", "author": [ { "family_name": "Buss", "given_name": "Samuel R.", "clpid": "Buss-S-R" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Pillay", "given_name": "Anand", "clpid": "Pillay-A" }, { "family_name": "Shore", "given_name": "Richard A.", "clpid": "Shore-R-A" } ], "abstract": "The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.", "doi": "10.2307/2687773", "issn": "1079-8986", "publisher": "Association for Symbolic Logic", "publication": "Bulletin of Symbolic Logic", "publication_date": "2001-06", "series_number": "2", "volume": "7", "issue": "2", "pages": "169-196" }, { "id": "authors:2q44f-edc44", "collection": "authors", "collection_id": "2q44f-edc44", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130610-142913085", "type": "article", "title": "Recent developments in the theory of Borel reducibility", "author": [ { "family_name": "Hjorth", "given_name": "Greg", "clpid": "Hjorth-G" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "Let E_0 be the Vitali equivalence relation and E_3 the product of countably\nmany copies of E_0. Two new dichotomy theorems for Borel equivalence relations are\nproved. First, for any Borel equivalence relation E that is (Borel) reducible to E_3, either\nE is reducible to E_0 or else E_3 is reducible to E. Second, if E is a Borel equivalence\nrelation induced by a Borel action of a closed subgroup of the infinite symmetric group\nthat admits an invariant metric, then either E is reducible to a countable Borel equivalence\nrelation or else E_3 is reducible to E.\nWe also survey a number of results and conjectures concerning the global structure\nof reducibility on Borel equivalence relations.", "issn": "0016-2736", "publisher": "Institute of Mathematics, Polish Academy of Sciences", "publication": "Fundamenta Mathematicae", "publication_date": "2001", "series_number": "1-2", "volume": "170", "issue": "1-2", "pages": "21-52" }, { "id": "authors:866an-x9143", "collection": "authors", "collection_id": "866an-x9143", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-095650096", "type": "article", "title": "Countable structures with a fixed group of automorphisms", "author": [ { "family_name": "Camerlo", "given_name": "Riccardo", "clpid": "Camerlo-R" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We prove that, given a countable group G, the set of countable structures (for a suitable language L)U_G whose automorphism group is isomorphic to G is a complete coanalytic set and if G \u2244 H then U_G is Borel inseparable from U_H . We give also a model theoretic interpretation of this result. We prove, in contrast, that the set of countable structures for L whose automorphism group is isomorphic to \u2124_p^\u2115 ,p a prime number, is \u03a0^1_1&\u03a3^1_1-complete.", "doi": "10.1007/BF02773566", "issn": "0021-2172", "publisher": "Hebrew University Magnes Press", "publication": "Israel Journal of Mathematics", "publication_date": "2000-12", "series_number": "1", "volume": "117", "issue": "1", "pages": "105-124" }, { "id": "authors:c39a4-aa811", "collection": "authors", "collection_id": "c39a4-aa811", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20111201-082818221", "type": "article", "title": "On the Classification Problem for Rank 2 Torsion-Free Abelian Groups", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We study here some foundational aspects of the classification problem for torsion-free abelian groups of finite rank. These are, up to isomorphism, the subgroups of the additive groups (Q^n, +), for some n = 1, 2, 3,.... The torsion-free abelian groups of rank \u2264 n are the subgroups of (Q^n, +).", "doi": "10.1112/S0024610700001411", "issn": "0024-6107", "publisher": "London Mathematical Society", "publication": "Journal of the London Mathematical Society", "publication_date": "2000-10", "series_number": "2", "volume": "62", "issue": "2", "pages": "437-450" }, { "id": "authors:wd197-d4m26", "collection": "authors", "collection_id": "wd197-d4m26", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130524-130949007", "type": "article", "title": "Linear algebraic groups and countable Borel equivalence relations", "author": [ { "family_name": "Adams", "given_name": "Scot", "clpid": "Adams-Scot" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "This paper is a contribution to the study of Borel equivalence relations on standard\nBorel spaces (i.e., Polish spaces equipped with their Borel structure). In\nmathematics one often deals with problems of classification of objects up to some\nnotion of equivalence by invariants. Frequently these objects can be viewed as elements\nof a standard Borel space X and the equivalence turns out to be a Borel\nequivalence relation E on X. A complete classification of X up to E consists of\nfinding a set of invariants I and a map c : X \u2192 I such that xEy \u21d4 c(x) = c(y).\nFor this to be of any interest both I and c must be explicit or definable and as simple\nand concrete as possible. The theory of Borel equivalence relations studies the\nset-theoretic nature of possible invariants and develops a mathematical framework\nfor measuring the complexity of such classification problems.", "doi": "10.1090/S0894-0347-00-00341-6", "issn": "0894-0347", "publisher": "American Mathematical Society", "publication": "Journal of the American Mathematical Society", "publication_date": "2000-06-23", "series_number": "4", "volume": "13", "issue": "4", "pages": "909-943" }, { "id": "authors:p9dz3-69j53", "collection": "authors", "collection_id": "p9dz3-69j53", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130731-144914781", "type": "book_section", "title": "Descriptive Dynamics", "book_title": "Descriptive set theory and dynamical systems", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Foreman", "given_name": "M.", "clpid": "Foreman-M" }, { "family_name": "Kechris", "given_name": "A. S.", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "A.", "clpid": "Louveau-A" }, { "family_name": "Weiss", "given_name": "B.", "clpid": "Weiss-B" } ], "abstract": "The purpose of the following informal lectures is to give a brief introduction to\ndescriptive dynamics, which I understand here to be the descriptive theory of\nPolish group actions. I will concentrate on the foundations, and hopefully at\na level accessible to anyone with a basic knowledge of descriptive set theory.\nI will illustrate some of the main methods used in this area, including Baire\ncategory arguments and various implementations of the \"changing the topology\"\ntechnique. A general reference for the results discussed in this paper is\nBecker-Kechris [1996].", "isbn": "9780521786447", "publisher": "Cambridge University Press", "place_of_publication": "New York", "publication_date": "2000-06", "pages": "231-258" }, { "id": "authors:absez-hgb80", "collection": "authors", "collection_id": "absez-hgb80", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-075724313", "type": "article", "title": "The complexity of the classification of Riemann surfaces and complex manifolds", "author": [ { "family_name": "Hjorth", "given_name": "G.", "clpid": "Hjorth-G" }, { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "In answer to a question by Becker, Rubel, and Henson, we show that countable subsets of \u2102 can be used as complete invariants for Riemann surfaces considered up to conformal equivalence, and that this equivalence relation is itself Borel in a natural Borel structure on the space of all such surfaces. We further proceed to precisely calculate the classification difficulty of this equivalence relation in terms of the modern theory of Borel equivalence relations. \n\nOn the other hand we show that the analog of Becker, Rubel, and Henson's question has a negative solution in (complex) dimension n \u2265 2.", "issn": "0019-2082", "publisher": "University of Illinois", "publication": "Illinois Journal of Mathematics", "publication_date": "2000-03", "series_number": "1", "volume": "44", "issue": "1", "pages": "104-137" }, { "id": "authors:pz5er-rys37", "collection": "authors", "collection_id": "pz5er-rys37", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130624-113819994", "type": "book_section", "title": "How Many Turing Degrees are There?", "book_title": "Computability theory and its applications: current trends and open problems", "author": [ { "family_name": "Dougherty", "given_name": "Randall", "clpid": "Dougherty-R" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Cholak", "given_name": "Peter", "clpid": "Cholak-P" } ], "abstract": "A Borel equivalence relation on a Polish space is countable if all\nof its equivalence classes are countable. Standard examples of countable Borel\nequivalence relations (on the space of subsets of the integers) that occur in\nrecursion theory are: recursive isomorphism, Turing equivalence, arithmetic\nequivalence, etc. There is a canonical hierarchy of complexity of countable\nBorel equivalence relations imposed by the notion of Borel reducibility. We\nwill survey results and conjectures concerning the problem of identifying the\nplace in this hierarchy of these equivalence relations from recursion theory and\nalso discuss some of their implications.", "doi": "10.1090/conm/257", "isbn": "9780821819227", "publisher": "American Mathematical Society", "place_of_publication": "Providence, RI", "publication_date": "2000", "pages": "83-94" }, { "id": "authors:yw7zg-jrn74", "collection": "authors", "collection_id": "yw7zg-jrn74", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-095020744", "type": "article", "title": "New Directions in Descriptive Set Theory", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "I will start with a quick definition of descriptive set theory: It is the study of the structure of definable sets and functions in separable completely metrizable spaces. Such spaces are usually called Polish spaces. Typical\nexamples are R^n, C^n, (separable) Hilbert space and more generally all separable Banach spaces, the Cantor space 2^N, the Baire space N^N, the infinite symmetric group S_\u221e, the unitary group (of the Hilbert space), the group of\nmeasure preserving transformations of the unit interval, etc.", "doi": "10.2307/421088", "issn": "1079-8986", "publisher": "Association for Symbolic Logic", "publication": "Bulletin of Symbolic Logic", "publication_date": "1999-06", "series_number": "2", "volume": "5", "issue": "2", "pages": "161-174" }, { "id": "authors:mbe97-y1b24", "collection": "authors", "collection_id": "mbe97-y1b24", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130517-100529842", "type": "article", "title": "Borel Chromatic Numbers", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Solecki", "given_name": "S.", "clpid": "Solecki-S" }, { "family_name": "Todorcevic", "given_name": "S.", "clpid": "Todorcevic-S" } ], "abstract": "We study in this paper graph coloring problems in the context of descriptive set theory. We consider graphs G=(X, R), where the vertex set X is a standard Borel space (i.e., a complete separable metrizable space equipped with its \u03c3-algebra of Borel sets), and the edge relation R \u2286 X^2 is\n\"definable\", i.e., Borel, analytic, co-analytic, etc.\n\nA Borel n-coloring of such a graph, where 1\u2a7d n \u2a7d N_0 , is a Borel map c: X \u2192 Y with card(Y)=n, such\nthat xRy\u21d2c(x) \u2260 {c( y). If such a Borel coloring exists we define the Borel\nchromatic number of G, in symbols X_B(G), to be the smallest such n.\nOtherwise we say that G has uncountable Borel chromatic number, in symbols X_B(G) > N_0.", "doi": "10.1006/aima.1998.1771", "issn": "0001-8708", "publisher": "Elsevier", "publication": "Advances in Mathematics", "publication_date": "1999-01-15", "series_number": "1", "volume": "141", "issue": "1", "pages": "1-44" }, { "id": "authors:jzzgb-w4761", "collection": "authors", "collection_id": "jzzgb-w4761", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130517-113148990", "type": "article", "title": "Rigidity properties of Borel ideals on the integers", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We classify the Borel ideals I on the set of natural numbers N for which p(N)/I can be Borel embedded into the orbit space of a Borel action of the infinite symmetric group. As a consequence we show that certain Borel ideals I, including the Fr\u00e9chet ideal, are completely characterized by the \"Borel cardinality\" of the set p(N)/I.", "doi": "10.1016/S0166-8641(97)00150-8", "issn": "0166-8641", "publisher": "Elsevier", "publication": "Topology and Its Applications", "publication_date": "1998-05-22", "series_number": "1-3", "volume": "85", "issue": "1-3", "pages": "195-205" }, { "id": "authors:pryh3-whk19", "collection": "authors", "collection_id": "pryh3-whk19", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130517-142221264", "type": "article", "title": "Borel equivalence relations induced by actions of the symmetric group", "author": [ { "family_name": "Hjorth", "given_name": "Greg", "clpid": "Hjorth-G" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "abstract": "We consider Borel equivalence relations E induced by actions of the infinite symmetric group, or equivalently the isomorphism relation on classes of countable models of bounded Scott rank. We relate the descriptive complexity of the equivalence relation to the nature of its complete invariants. A typical theorem is that E is potentially \u03a0^0_3 iff the invariants are countable sets of reals, it is potentially \u03a0^0_4 iff the invariants are countable sets of countable sets of reals, and so on. The proofs use various techniques, including Vaught transforms, changing topologies, and the Scott analysis of countable models.", "doi": "10.1016/S0168-0072(97)00049-3", "issn": "0168-0072", "publisher": "Elsevier", "publication": "Annals of Pure and Applied Logic", "publication_date": "1998-03-11", "series_number": "1", "volume": "92", "issue": "1", "pages": "63-112" }, { "id": "authors:tjnvz-v5872", "collection": "authors", "collection_id": "tjnvz-v5872", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-105907286", "type": "article", "title": "Measurable enumeration of eigenelements", "author": [ { "family_name": "Gordon", "given_name": "Alexander Y.", "clpid": "Gordon-A-Y" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We prove that for eigenelements of a measurable family of linear self-adjoint operators in a separable Hilbert space there exists a measurable enumeration. We also prove a similar result for measurable families of bounded linear operators having at most countably many eigenvalues (under certain restrictions on the parameter space). The proof of the latter result is based on descriptive set theory, while in the case of self-adjoint (and some more general) operators the proof is constructive.", "issn": "0003-6811", "publisher": "Taylor and Francis", "publication": "Applicable Analysis: An International Journal", "publication_date": "1998", "series_number": "1-4", "volume": "71", "issue": "1-4", "pages": "41-61" }, { "id": "authors:71r71-7n935", "collection": "authors", "collection_id": "71r71-7n935", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-083928827", "type": "book_section", "title": "The descriptive classification of some classes of C\u2217-algebras", "book_title": "Proceedings of the Sixth Asian Logic Conference", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Chong", "given_name": "C. -T.", "clpid": "Chong-C-T" } ], "abstract": "We introduce here a parametrization of separable C*-algebras by a\nstandard Borel space and study the descriptive complexity of various canonical\nclasses of C*-algebras in this parametrization. This can be viewed as\nproviding an analog of the corresponding classification of classes of von Neumann\nalgebras (acting on a fixed separable Hilbert space) in the Effros Borel\nspace of von Neumann algebras (see for example Nielsen [13]). However, in\ncontrast with the von Neumann case, where most interesting classes (like:\nfactors, type I, II, III, hyperfinite) turn out to be Borel, in the C*-algebra\ncase many important classes turn out to be co-analytic but not Borel. This\nmakes the situation more interesting from the set-theoretic point of view,\nand leads to further questions, like the construction of canonical co-analytic\nnorms, which we also address here. Finally, we relate the above to work of\nSutherland [17] on parametrization of Polish groups and indicate how one\nmight be able to show that various familiar classes of (second countable)\nlocally compact groups are non-Borel as well.", "isbn": "9789810234324", "publisher": "World Scientific", "place_of_publication": "River Edge, NJ", "publication_date": "1998", "pages": "121-149" }, { "id": "authors:2c6jq-p1053", "collection": "authors", "collection_id": "2c6jq-p1053", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130520-144530065", "type": "article", "title": "New Dichotomies for Borel Equivalence Relations", "author": [ { "family_name": "Hjorth", "given_name": "Greg", "clpid": "Hjorth-G" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We announce two new dichotomy theorems for Borel equivalence relations, and present the results in context by giving an overview of related recent developments.", "doi": "10.2307/421148", "issn": "1079-8986", "publisher": "Association for Symbolic Logic", "publication": "Bulletin of Symbolic Logic", "publication_date": "1997-09", "series_number": "3", "volume": "3", "issue": "3", "pages": "329-346" }, { "id": "authors:fzwwc-ewv74", "collection": "authors", "collection_id": "fzwwc-ewv74", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-135837396", "type": "article", "title": "On the concept of \u03c0^1_1-completeness", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "It is shown that two natural notions of completeness for co-analytic sets in Polish spaces, one in terms of continuous reductions and the other in terms of Borel reductions, coincide. The proof uses methods of effective descriptive set theory.", "doi": "10.1090/S0002-9939-97-03770-2", "issn": "0002-9939", "publisher": "American Mathematical Society", "publication": "Proceedings of the American Mathematical Society", "publication_date": "1997-06", "series_number": "6", "volume": "125", "issue": "6", "pages": "1811-1814" }, { "id": "authors:56k1v-f9h12", "collection": "authors", "collection_id": "56k1v-f9h12", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-133826211", "type": "article", "title": "The classification of hypersmooth Borel equivalence relations", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "abstract": "This paper is a contribution to the study of Borel equivalence relations in standard Borel spaces, i.e., Polish spaces equipped with their Borel structure. A class of such equivalence relations which has received particular attention is the class of hyperfinite Borel equivalence relations. These can be defined as the increasing unions of sequences of Borel equivalence relations all of whose equivalence classes are finite or, as it turns out, equivalently those induced by the orbits of a single Borel auto-morphism. Hyperfinite equivalence relations have been classified in [DJK], under two notions of equivalence, Borel bi-reducibility, and Borel isomorphism.", "doi": "10.1090/S0894-0347-97-00221-X", "issn": "0894-0347", "publisher": "American Mathematical Society", "publication": "Journal of the American Mathematical Society", "publication_date": "1997-01", "series_number": "1", "volume": "10", "issue": "1", "pages": "215-242" }, { "id": "authors:mnx01-0pa63", "collection": "authors", "collection_id": "mnx01-0pa63", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130517-141337004", "type": "article", "title": "Borel equivalence relations and classifications of countable models", "author": [ { "family_name": "Hjorth", "given_name": "Greg", "clpid": "Hjorth-G" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "Using the theory of Borel equivalence relations we analyze the isomorphism relation on the countable models of a theory and develop a framework for measuring the complexity of possible complete invariants for isomorphism.", "doi": "10.1016/S0168-0072(96)00006-1", "issn": "0168-0072", "publisher": "Elsevier", "publication": "Annals of Pure and Applied Logic", "publication_date": "1996-12-15", "series_number": "3", "volume": "82", "issue": "3", "pages": "221-272" }, { "id": "authors:tvcy4-gz126", "collection": "authors", "collection_id": "tvcy4-gz126", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180809-161147539", "type": "book_section", "title": "Better topologies", "book_title": "The Descriptive Set Theory of Polish Group Actions", "author": [ { "family_name": "Becker", "given_name": "Howard", "clpid": "Becker-H-S" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "This section of the book contains several results, all of which are about changing the topology on a G\u2013space X. Given an action a : G \u00d7 X \u2192 X by a Polish group G, we would like to put a new topology t on X such that: (X,t) is \"nice\", e.g., metrizable or Polish; a is continuous with respect to t; and some given a\u2013invariant set I \u2286 X is t\u2013open (or closed or clopen). We have two different types of theorems, one type which says that there is a new topology which makes the action continuous, the other type which says that given a continuous action, there is a new, finer topology in which the action is still continuous and which makes an invariant set open. We have already seen an instance of such a result in 2.6.6, from which it follows that if G is Polish and X a Borel G\u2013space, then we can put on X a separable metrizable topology having the same Borel structure in which the action is continuous. \n\nIn \u00a75.1, 5.2 we consider the most important case, that of Borel\u2013measurable actions and Borel invariant sets. In \u00a75.3, 5.4 we consider more general actions and invariant sets.", "doi": "10.1017/cbo9780511735264.008", "isbn": "9780511735264", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "1996", "pages": "53-81" }, { "id": "authors:6aem8-0j159", "collection": "authors", "collection_id": "6aem8-0j159", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180816-134913115", "type": "book_section", "title": "Polish Groups", "book_title": "The Descriptive Set Theory of Polish Group Actions", "author": [ { "family_name": "Becker", "given_name": "Howard", "clpid": "Becker-H-S" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Becker", "given_name": "Howard", "clpid": "Becker-H-S" }, { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "abstract": "We will survey here some basic facts from the theory of Polish groups.", "doi": "10.1017/CBO9780511735264.004", "isbn": "9780521576055", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "1996", "pages": "3-12" }, { "id": "authors:n9n01-6p391", "collection": "authors", "collection_id": "n9n01-6p391", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-154314089", "type": "book_section", "title": "Actions of Polish Groups", "book_title": "The Descriptive Set Theory of Polish Group Actions", "author": [ { "family_name": "Becker", "given_name": "Howard", "clpid": "Becker-H" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Becker", "given_name": "Howard", "clpid": "Becker-H" }, { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "abstract": "In \u00a72.1 - 2.5, we survey some basic facts about Polish group actions, particularly continuous and Borel measurable actions on Polish spaces and standard Borel spaces, respectively. In \u00a72.6 we establish the existence of universal actions for any Polish group, extending a result of Mackey and Varadarajan from the locally compact case. Finally, \u00a72.7 is concerned with universal actions for S\u221e and its closed subgroups.", "doi": "10.1017/CBO9780511735264.005", "isbn": "9780521576055", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "1996", "pages": "13-32" }, { "id": "authors:2gx5k-zp413", "collection": "authors", "collection_id": "2gx5k-zp413", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-094122252", "type": "book_section", "title": "Introduction", "book_title": "The Descriptive Set Theory of Polish Group Actions", "author": [ { "family_name": "Becker", "given_name": "Howard", "clpid": "Becker-H" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Becker", "given_name": "Howard", "clpid": "Becker-H" }, { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "abstract": "A Polish space (group) is a separable, completely metrizable topological space (group). This book is about actions of Polish groups, in connection with \u2013 or from the point of view of \u2013 the subject of descriptive set theory. Descriptive set theory is the study of definable sets and functions in Polish spaces. The basic classes of definable sets are the classes of Borel, analytic and co\u2013analytic sets, and these certainly constitute the main topic of the book, but at times we also consider other classes of definable sets.\n\nThe structure of Borel actions of Polish locally compact, i.e., second countable locally compact, topological groups has long been studied in ergodic theory, operator algebras and group representation theory. See, for example, Auslander\u2013Moore [66], Feldman\u2013Hahn\u2013Moore [78], Glimm [61], Kechris [92a], Mackey [57, 62, 89], Moore [82], Ramsay [82, 85], Sinai [89], Varadarajan [63], Vershik\u2013Fedorov [87], Zimmer [84] for a sample of this work. This is closely related to the subject matter of this book. More recently, there has been increasing interest in an extension of the above: studying the structure of Borel actions of arbitrary, not necessarily locally compact, Polish groups.", "doi": "10.1017/CBO9780511735264.002", "isbn": "9780521576055", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "1996", "pages": "viii-xii" }, { "id": "authors:2eyc0-mcp07", "collection": "authors", "collection_id": "2eyc0-mcp07", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180815-091513340", "type": "book", "title": "The Descriptive Set Theory of Polish Group Actions", "author": [ { "family_name": "Becker", "given_name": "Howard" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Becker", "given_name": "Howard", "clpid": "Becker-H-S" }, { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "abstract": "In this book the authors present their research into the foundations of the theory of Polish groups and the associated orbit equivalence relations. The particular case of locally compact groups has long been studied in many areas of mathematics. Non-locally compact Polish groups occur naturally as groups of symmetries in such areas as logic (especially model theory), ergodic theory, group representations, and operator algebras. Some of the topics covered here are: topological realizations of Borel measurable actions; universal actions; applications to invariant measures; actions of the infinite symmetric group in connection with model theory (logic actions); dichotomies for orbit spaces (including Silver, Glimm-Effros type dichotomies and the topological Vaught conjecture); descriptive complexity of orbit equivalence relations; definable cardinality of orbit spaces.", "doi": "10.1017/CBO9780511735264", "isbn": "9780521576055", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "1996" }, { "id": "authors:5qe0p-g3v80", "collection": "authors", "collection_id": "5qe0p-g3v80", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-083346365", "type": "book_section", "title": "Actions with Borel Orbit Equivalence Relations", "book_title": "The Descriptive Set Theory of Polish Group Actions", "author": [ { "family_name": "Becker", "given_name": "Howard", "clpid": "Becker-H" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Becker", "given_name": "Howard", "clpid": "Becker-H" }, { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "abstract": "If G is a Polish group and (X, \u03b1) a Borel G-space, then the associated equivalence relation E_\u03b1 is analytic but in general not Borel, even when X is a Polish G-space (see 3.2). We study in this section various questions concerning the Borelness of such E_\u03b1.", "doi": "10.1017/CBO9780511735264.010", "isbn": "9780521576055", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "1996", "pages": "98-115" }, { "id": "authors:brfgs-28861", "collection": "authors", "collection_id": "brfgs-28861", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-132523681", "type": "book_section", "title": "Preface", "book_title": "The Descriptive Set Theory of Polish Group Actions", "author": [ { "family_name": "Becker", "given_name": "Howard", "clpid": "Becker-H" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Becker", "given_name": "Howard", "clpid": "Becker-H" }, { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "abstract": "This book contains the results of the research of the authors in the foundations of the theory of definable actions of Polish groups and the associated orbit equivalence relations. It is addressed to graduate students and researchers in this area as well as to mathematicians in other fields, whose work comes in contact with this subject.\n\nThe reader will also find some new theorems which are attributed to someone other than the authors. Many people have helped in one way or another to improve this book: by allowing us to include these new theorems; by simplifying some of our proofs; by bringing to our attention some work we were unaware of; or by reading earlier versions of this book \u2013 or parts of it \u2013 and suggesting improvements in the exposition. We thank all of them: G. Cherlin, W. Comfort, R. Dougherty, G. Hjorth, A. Louveau, M. Megrelishvili, M. G. Nadkarni, R. Sami, S. Solecki, S. Thomas, and an anonymous referee.", "doi": "10.1017/CBO9780511735264.001", "isbn": "9780521576055", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "1996", "pages": "vii" }, { "id": "authors:c84aj-kqp16", "collection": "authors", "collection_id": "c84aj-kqp16", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180816-125320670", "type": "book_section", "title": "Model Theory and the Vaught Conjecture", "book_title": "The Descriptive Set Theory of Polish Group Actions", "author": [ { "family_name": "Becker", "given_name": "Howard", "clpid": "Becker-H-S" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Becker", "given_name": "Howard", "clpid": "Becker-H-S" }, { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "abstract": "The Vaught Conjecture is a notorious open problem in mathematical logic. A number of stronger conjectures have subsequently been proposed, some of which are conjectures about the descriptive set theory of Polish group actions. These later conjectures are known (collectively) as the Topological Vaught Conjecture. In \u00a76.1 we give some background on the Vaught Conjecture. This section is written primarily for non-logicians, and contains\nnothing that is not well known to specialists. \u00a76.2 contains a number of results, some new, about the Topological Vaught Conjecture. Our new results are of two types: First, that various conjectures imply - or are equivalent to - various other conjectures, and, second, that certain even stronger conjectures are false. There are few things to be proved in \u00a76.2; the results here are mostly corollaries of theorems in earlier parts of this book. In \u00a76.3 we prove a theorem about atomic models (which has nothing to do with the Vaught Conjecture). More connections between group actions and logic will be discussed in \u00a77.", "doi": "10.1017/CBO9780511735264.009", "isbn": "9780521576055", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "1996", "pages": "82-97" }, { "id": "authors:tefar-hsx94", "collection": "authors", "collection_id": "tefar-hsx94", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180821-074806630", "type": "book_section", "title": "Invariant Measures and Paradoxical Decompositions", "book_title": "The Descriptive Set Theory of Polish Group Actions", "author": [ { "family_name": "Becker", "given_name": "Howard", "clpid": "Becker-H" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Becker", "given_name": "Howard", "clpid": "Becker-H" }, { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" } ], "abstract": "We will combine here a theorem of Nadkarni [90] with the results proved in \u00a72 to obtain an application to the theory of paradoxical decompositions. Our general reference for this subject is Wagon [93]. The results in \u00a74 are not used in the rest of the book.", "doi": "10.1017/CBO9780511735264.007", "isbn": "9780521576055", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "1996", "pages": "44-52" }, { "id": "authors:krf6r-wtm04", "collection": "authors", "collection_id": "krf6r-wtm04", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130520-160211996", "type": "article", "title": "Analytic Equivalence Relations and Ulm-Type Classifications", "author": [ { "family_name": "Hjorth", "given_name": "Greg", "clpid": "Hjorth-G" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "Our main goal in this paper is to establish a Glimm-Effros type dichotomy for arbitrary analytic equivalence relations.", "doi": "10.2307/2275888", "issn": "0022-4812", "publisher": "Association for Symbolic Logic", "publication": "Journal of Symbolic Logic", "publication_date": "1995-12", "series_number": "4", "volume": "60", "issue": "4", "pages": "1273-1300" }, { "id": "authors:bjnyy-epy71", "collection": "authors", "collection_id": "bjnyy-epy71", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-103730289", "type": "article", "title": "Approximation of analytic by Borel sets and definable countable chain conditions", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Solecki", "given_name": "S.", "clpid": "Solecki-S" } ], "abstract": "Let I be a \u03c3-ideal on a Polish space such that each set from I is contained in a Borel set from I. We say that I fails to fulfil the \u03a3_1^1 countable chain condition if there is a \u03a3_1^1 equivalence relation with uncountably many equivalence classes none of which is in I. Assuming definable determinacy, we show that if the family of Borel sets from I is definable in the codes of Borel sets, then each \u03a3_1^1 set is equal to a Borel set modulo a set from I iff I fulfils the \u03a3_1^1 countable chain condition. Further we characterize the \u03c3-ideals I generated by closed sets that satisfy the countable chain condition or, equivalently in this case, the approximation property for \u03a3_1^1 sets mentioned above. It turns out that they are exactly of the form MGR(F)={A : \u2200F \u2208 F A \u2229F is meager in F} for a countable family F of closed sets. In particular, we verify partially a conjecture of Kunen by showing that the \u03c3-ideal of meager sets is the unique \u03c3-ideal on R, or any Polish group, generated by closed sets which is invariant under translations and satisfies the countable chain condition.", "doi": "10.1007/BF02808208", "issn": "0021-2172", "publisher": "Hebrew University Magnes Press", "publication": "Israel Journal of Mathematics", "publication_date": "1995-10", "series_number": "1-3", "volume": "89", "issue": "1-3", "pages": "343-356" }, { "id": "authors:p3p13-xp104", "collection": "authors", "collection_id": "p3p13-xp104", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130517-110331634", "type": "article", "title": "Topology and descriptive set theory", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "This paper consists essentially of the text of a series of four lectures given by the author in the Summer Conference on General Topology and Applications, Amsterdam, August 1994. \nInstead of attempting to give a general survey of the interrelationships between the two subjects mentioned in the title, which would be an enormous and hopeless task, we chose to illustrate them in a specific context, that of the study of Borel actions of Polish groups and Borel equivalence relations. This is a rapidly growing area of research of much current interest, which has interesting connections not only with topology and set theory (which are emphasized here), but also to ergodic theory, group representations, operator algebras and logic (particularly model theory and recursion theory).\nThere are four parts, corresponding roughly to each one of the lectures. The first contains a brief review of some fundamental facts from descriptive set theory. In the second we discuss Polish groups, and in the third the basic theory of their Borel actions. The last part concentrates on Borel equivalence relations.\nThe exposition is essentially self-contained, but proofs, when included at all, are often given in the barest outline.", "doi": "10.1016/0166-8641(94)00146-4", "issn": "0166-8641", "publisher": "Elsevier", "publication": "Topology and Its Applications", "publication_date": "1994-08-15", "series_number": "3", "volume": "58", "issue": "3", "pages": "195-222" }, { "id": "authors:r3tcq-5nk56", "collection": "authors", "collection_id": "r3tcq-5nk56", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-092754518", "type": "article", "title": "The structure of hyperfinite Borel equivalence relations", "author": [ { "family_name": "Dougherty", "given_name": "R.", "clpid": "Dougherty-R" }, { "family_name": "Jackson", "given_name": "S.", "clpid": "Jackson-S" }, { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We study the structure of the equivalence relations induced by the orbits of a single Borel automorphism on a standard Borel space. We show that any two such equivalence relations which are not smooth, i.e., do not admit Borel selectors, are Borel embeddable into each other. (This utilizes among other things work of Effros and Weiss.) Using this and also results of Dye, Varadarajan, and recent work of Nadkarni, we show that the cardinality of the set of ergodic invariant measures is a complete invariant for Borel isomorphism of aperiodic nonsmooth such equivalence relations. In particular, since the only possible such cardinalities are the finite ones, countable infinity, and the cardinality of the continuum, there are exactly countably infinitely many isomorphism types. Canonical examples of each type are also discussed.", "doi": "10.2307/2154620", "issn": "0002-9947", "publisher": "American Mathematical Society", "publication": "Transactions of the American Mathematical Society", "publication_date": "1994-01", "series_number": "1", "volume": "341", "issue": "1", "pages": "193-225" }, { "id": "authors:2whnz-a9y96", "collection": "authors", "collection_id": "2whnz-a9y96", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130524-135334590", "type": "article", "title": "Countable Sections for Locally Compact Group Actions. II", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "In this paper we study the structure of the orbit equivalence relation induced by a Borel action of a second countable locally compact group on a standard Borel space.", "issn": "0002-9939", "publisher": "American Mathematical Society", "publication": "Proceedings of the American Mathematical Society", "publication_date": "1994-01", "series_number": "1", "volume": "120", "issue": "1", "pages": "241-247" }, { "id": "authors:hgkgt-0dm64", "collection": "authors", "collection_id": "hgkgt-0dm64", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-103113640", "type": "article", "title": "Amenable Versus Hyperfinite Borel Equivalence Relations", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "Let X be a standard Borel space (i.e., a Polish space with the associated Borel structure), and let E be a countable Borel equivalence relation on X, i.e., a Borel equivalence relation E for which every equivalence class [X]_E is countable. By a result of Feldman-Moore [FM], E is induced by the orbits of a Borel action of a countable group G on X.", "doi": "10.2307/2275102", "issn": "0022-4812", "publisher": "Association for Symbolic Logic", "publication": "Journal of Symbolic Logic", "publication_date": "1993-09", "series_number": "3", "volume": "58", "issue": "3", "pages": "894-907" }, { "id": "authors:raqc2-nra14", "collection": "authors", "collection_id": "raqc2-nra14", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130523-095229531", "type": "article", "title": "Borel actions of Polish groups", "author": [ { "family_name": "Becker", "given_name": "Howard", "clpid": "Becker-H" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We show that a Borel action of a Polish group on a standard Borel space is Borel isomorphic to a continuous action of the group on a Polish space, and we apply this result to three aspects of the theory of Borel actions of Polish groups: universal actions, invariant probability measures and the topological Vaught conjecture. We establish the existence of universal actions for any given Polish group, extending a result of Mackey and Varadarajan for the locally compact case. We prove an analogue of Tarski's theorem on paradoxical decompositions, by showing that the existence of an invariant Borel probability measure is equivalent to the nonexistence of paradoxical decompositions with countably many Borel pieces. We show that various natural versions of the topological Vaught conjecture are equivalent to each other and, in the case of the group of permutations of N, with the model-theoretic Vaught conjecture for infinitary logic; this depends on our identification of the universal action for that group.", "doi": "10.1090/S0273-0979-1993-00383-5", "issn": "0273-0979", "publisher": "American Mathematical Society", "publication": "Bulletin of the American Mathematical Society", "publication_date": "1993-04", "series_number": "2", "volume": "28", "issue": "2", "pages": "334-341" }, { "id": "authors:1b42s-01575", "collection": "authors", "collection_id": "1b42s-01575", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130516-154918991", "type": "article", "title": "Countable sections for locally compact group actions", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "It has been shown by J. Feldman, P. Hahn and C. C. Moore that every\nnon-singular action of a second countable locally compact group has a countable\n(in fact so-called lacunary) complete measurable section. This is extended here to\nthe purely Borel theoretic category, consisting of a Borel action of such a group on\nan analytic Borel space (without any measure). Characterizations of when an\narbitrary Borel equivalence relation admits a countable complete Borel section are\nalso established.", "doi": "10.1017/S0143385700006751", "issn": "0143-3857", "publisher": "Cambridge University Press", "publication": "Ergodic Theory and Dynamical Systems", "publication_date": "1992-06-01", "series_number": "2", "volume": "12", "issue": "2", "pages": "283-295" }, { "id": "authors:0kbz6-x3q75", "collection": "authors", "collection_id": "0kbz6-x3q75", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-082635644", "type": "article", "title": "Descriptive Set Theory and Harmonic Analysis", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "A.", "clpid": "Louveau-A" } ], "abstract": "During the 1989 European ASL Summer Meeting in Berlin, the\nauthors gave a series of eight lectures (short course) on the topic of the title. This\nsurvey article consists basically of the lecture notes for that course distributed to\nthe participants of that conference. We have purposely tried in this printed version\nto preserve the informal style of the original notes.", "doi": "10.2307/2275277", "issn": "0022-4812", "publisher": "Association for Symbolic Logic", "publication": "Journal of Symbolic Logic", "publication_date": "1992-06", "series_number": "2", "volume": "57", "issue": "2", "pages": "413-441" }, { "id": "authors:1np55-jtn67", "collection": "authors", "collection_id": "1np55-jtn67", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130516-154155660", "type": "book_section", "title": "The structure of Borel equivalence relations in Polish spaces", "book_title": "Set theory of the continuum", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Judah", "given_name": "Haim", "clpid": "Judah-H" }, { "family_name": "Just", "given_name": "Winfried", "clpid": "Just-W" }, { "family_name": "Woodin", "given_name": "Hugh", "clpid": "Woodin-H" } ], "abstract": "An exposition of recent work on Borel equivalence relations in Polish spaces is presented. This includes a general Glimm-Effros dichotomy for Borel equivalence relations and a study of countable Borel equivalence relations and their classification into subclasses such as smooth, hyperfinite, amenable, treeable etc.", "doi": "10.1007/978-1-4613-9754-0_7", "isbn": "978-1-4613-9756-4", "publisher": "Springer-Verlag", "place_of_publication": "New York", "publication_date": "1992", "pages": "89-102" }, { "id": "authors:bewz5-6x728", "collection": "authors", "collection_id": "bewz5-6x728", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130517-102333066", "type": "article", "title": "The Complexity of Antidifferentiation", "author": [ { "family_name": "Dougherty", "given_name": "Randall", "clpid": "Dougherty-R" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We consider real-valued functions defined on the interval [0, 1]. We denote by \u0394 the set of derivatives; i.e., \u0192 \u0404 \u0394 iff there is a differentiable function F such that F' = \u0192. Any such F is a primitive of \u0192 and is uniquely determined up to a constant. To normalize, we denote by F(x)=\u0192^x_0\u0192 the\nprimitive determined by F(0)=0. This is the original Newtonian concept of integration as antidifferentiation.", "doi": "10.1016/0001-8708(91)90006-S", "issn": "0001-8708", "publisher": "Elsevier", "publication": "Advances in Mathematics", "publication_date": "1991-08", "series_number": "2", "volume": "88", "issue": "2", "pages": "145-169" }, { "id": "authors:q6sdf-63e75", "collection": "authors", "collection_id": "q6sdf-63e75", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-093201219", "type": "article", "title": "Hereditary properties of the class of closed sets of uniqueness for trigonometric series", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "It is shown that the\u03c3-ideal U_0 of closed sets of extended uniqueness in T is hereditarily non-Borel, i.e. every \"non-trivial\" \u03c3-ideal of closed sets I \u2286 U_0 is non-Borel. This implies both the result of Solovay, Kaufman that both U_0 and U(the \u03c3-ideal of closed sets of uniqueness) are not Borel as well as the theorem of Debs-Saint Raymond that every Borel subset of T of extended uniqueness is of the first category. A further extension to ideals contained in U_0 is given.", "doi": "10.1007/BF02772948", "issn": "0021-2172", "publisher": "Hebrew University Magnes Press", "publication": "Israel Journal of Mathematics", "publication_date": "1991-06", "series_number": "2", "volume": "73", "issue": "2", "pages": "189-198" }, { "id": "authors:3b1ft-4qg60", "collection": "authors", "collection_id": "3b1ft-4qg60", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130517-105346791", "type": "article", "title": "A strong boundedness theorem for dilators", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Woodin", "given_name": "W. H.", "clpid": "Woodin-W-H" } ], "abstract": "We prove a strong boundedness theorem for dilators: if A \u2286 DIL is \u03a3^1_1, then there is a recursive dilator D_0 such that \u2200D \u2208 A (D can be embedded into D_0).", "doi": "10.1016/0168-0072(91)90041-J", "issn": "0168-0072", "publisher": "Elsevier Masson", "publication": "Annals of Pure and Applied Logic", "publication_date": "1991-04-15", "series_number": "1-2", "volume": "52", "issue": "1-2", "pages": "93-97" }, { "id": "authors:sfhk2-sa892", "collection": "authors", "collection_id": "sfhk2-sa892", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130517-103738099", "type": "article", "title": "Boundedness theorems for dilators and ptykes", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "The main theorem of this paper is: If \u0192 is a partial function from \u2135_1 to \u2135_1 which is \u2211^1_1-bounded, then there is a weakly finite primitive recursive dilator D such that for all infinite \u03b1 \u03f5 dom(\u0192), \u0192(\u03b1) \u2a7d D(\u03b1). The proof involves only elementary combinatorial constructions of trees. A generalization to ptykes is also given.", "doi": "10.1016/0168-0072(91)90040-S", "issn": "0168-0072", "publisher": "Elsevier Masson", "publication": "Annals of Pure and Applied Logic", "publication_date": "1991-04-15", "series_number": "1-2", "volume": "52", "issue": "1-2", "pages": "79-92" }, { "id": "authors:rc9pp-sqd71", "collection": "authors", "collection_id": "rc9pp-sqd71", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-150609710", "type": "article", "title": "The Class Of Synthesizable Pseudomeasures", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "A.", "clpid": "Louveau-A" }, { "family_name": "Tardivel", "given_name": "V.", "clpid": "Tardivel-V" } ], "abstract": "In this paper we study descriptive set theoretic questions related to\nconcepts of harmonic synthesis on the unit circle T, and their relationship\nwith the structure of uniqueness sets.", "issn": "0019-2082", "publisher": "Illinois Journal of Mathematics", "publication": "Illinois Journal of Mathematics", "publication_date": "1991-03", "series_number": "1", "volume": "35", "issue": "1", "pages": "107-146" }, { "id": "authors:q6c4c-5ar39", "collection": "authors", "collection_id": "q6c4c-5ar39", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-150404346", "type": "article", "title": "Amenable Equivalence Relations and Turing Degrees", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "In [12] Slaman and Steel posed the following problem:\nAssume ZF + DC + AD. Suppose we have a function assigning to each Turing\ndegree d a linear order <_d of d. Then must the rationals embed order preservingly in\n<_d for a cone of d's?", "doi": "10.2307/2274913", "issn": "0022-4812", "publisher": "Association for Symbolic Logic", "publication": "Journal of Symbolic Logic", "publication_date": "1991-03", "series_number": "1", "volume": "56", "issue": "1", "pages": "182-194" }, { "id": "authors:cvb33-qt212", "collection": "authors", "collection_id": "cvb33-qt212", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-131913642", "type": "article", "title": "A Glimm-Effros dichotomy for Borel equivalence relations", "author": [ { "family_name": "Harrington", "given_name": "L. A.", "clpid": "Harrington-L-A" }, { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "A.", "clpid": "Louveau-A" } ], "abstract": "A basic dichotomy concerning the structure of the orbit space of a transformation group has been discovered by Glimm [G12] in the locally compact group action case and extended by Effros [E 1, E2] in the Polish group action\ncase when additionally the induced equivalence relation is F\u03c3. It is the purpose of this paper to extend the Glimm-Effros dichotomy to the very general context of an arbitrary Borel equivalence relation (not even necessarily induced by a group action). Despite the totally classical descriptive set-theoretic nature of our result, our proof requires the employment of methods of effective descriptive\nset theory and thus ultimately makes crucial use of computability (or recursion) theory on the integers.", "doi": "10.1090/S0894-0347-1990-1057041-5", "issn": "0894-0347", "publisher": "American Mathematical Society", "publication": "Journal of the American Mathematical Society", "publication_date": "1990-10", "series_number": "4", "volume": "3", "issue": "4", "pages": "903-928" }, { "id": "authors:xtm8s-6zw43", "collection": "authors", "collection_id": "xtm8s-6zw43", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-151355415", "type": "article", "title": "A Classification of Baire Class 1 Functions", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "A.", "clpid": "Louveau-A" } ], "abstract": "We study in this paper various ordinal ranks of (bounded) Baire class 1 functions and we show their essential equivalence. This leads to a natural classification of the class of bounded Baire class 1 functions B_1 in a transfinite hierarchy B^\u03be_1 \u03be < \u03c9_1) of \"small\" Baire classes, for which (for example) an analysis similar to the Hausdorff-Kuratowski analysis of \u0394^0_2 sets via transfinite differences of closed sets can be carried out. The notions of pseudouniform convergence of a sequence of functions and optimal convergence of a sequence of continuous functions to a Baire class 1 function \u0192 are introduced and used in this study.", "issn": "0002-9947", "publisher": "American Mathematical Society", "publication": "Transactions of the American Mathematical Society", "publication_date": "1990-03", "series_number": "1", "volume": "318", "issue": "1", "pages": "209-236" }, { "id": "authors:ryk0a-81x05", "collection": "authors", "collection_id": "ryk0a-81x05", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130612-084032401", "type": "article", "title": "Covering theorems for uniqueness and extended uniqueness sets", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "abstract": "A covering theorem for a class of sets C asserts that every set in C can be covered by a countable union of sets in some (somehow simpler) class C. In the theory of sets of uniqueness on the unit circle T the first result of this\nkind is Piatetski-Shapiro's theorem in [PS], which states that every closed set of uniqueness can be covered by countably many closed sets in the class U'_1, consisting of those closed sets E \u2286 T for which there exists a sequence of\nfunctions in A(T), vanishing on E, which converges to the function 1 in the weak^*-topology.", "issn": "0010-1354", "publisher": "Polish Academy of Sciences, Institute of Mathematics", "publication": "Colloquium Mathematicum", "publication_date": "1990", "series_number": "1", "volume": "59", "issue": "1", "pages": "63-79" }, { "id": "authors:akb20-gxm28", "collection": "authors", "collection_id": "akb20-gxm28", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130626-124718433", "type": "book_section", "title": "The Classification of Hyperfinite Borel Equivalence Relations", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Choquet", "given_name": "G.", "clpid": "Choquet-G" }, { "family_name": "Godefroy", "given_name": "G.", "clpid": "Godefroy-G" }, { "family_name": "Rogalski", "given_name": "M.", "clpid": "Rogalski-M" }, { "family_name": "Saint Raymond", "given_name": "J.", "clpid": "Saint-Raymond-J" } ], "abstract": "Let X be a standard Borel space and E a Borel equivalence relation on X. We call E hyperfinite if there is a Borel automorphism T of X such that xEy \u21d4 \u2203 n \u0404 \u2124(T^nx = y).\nFor Borel equivalence relations E,F on X, Y resp. we write\nE \u2291 F \u21d4 3 \u0192 : X \u2192 Y(\u0192 Borel, injective with E = \u0192^(-1)[F]) \nE \u2248 F \u21d4 E \u2291 F and F \u2291 E E \u2245 F \u21d4 \u2203 \u0192 :X \u2192 Y(\u0192 a Borel isomorphism with E= \u0192^(-1)[F]) A Borel equivalence relation E on X is called smooth if there is a Borel map \u0192: X \u2192 Y (Y\nsome standard Borel space) with xEy \u21d4 \u0192(x) = \u0192(y).", "publisher": "Universit\u00e9 Pierre et Marie Curie", "publication_date": "1990" }, { "id": "authors:x6qt8-jqp29", "collection": "authors", "collection_id": "x6qt8-jqp29", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-084327264", "type": "article", "title": "\u03a0^1_1 Borel Sets", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Marker", "given_name": "David", "clpid": "Marker-D" }, { "family_name": "Sami", "given_name": "Ramez L.", "clpid": "Sami-R-L" } ], "abstract": "The results in this paper were motivated by the following\nquestion of Sacks. Suppose T is a recursive theory with countably many countable models. What can you say about the least ordinal \u0251 such that all models of T have Scott rank below \u0251? If Martin's conjecture is true for T then \u0251 \u2264 \u03c9\u00b72.", "issn": "0022-4812", "publisher": "Association for Symbolic Logic", "publication": "Journal of Symbolic Logic", "publication_date": "1989-09", "series_number": "3", "volume": "54", "issue": "3", "pages": "915-920" }, { "id": "authors:wwr0g-prk26", "collection": "authors", "collection_id": "wwr0g-prk26", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130524-113257744", "type": "article", "title": "Hausdorff Measures and Sets of Uniqueness for Trigonometric Series", "author": [ { "family_name": "Dougherty", "given_name": "R.", "clpid": "Dougherty-R" }, { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We characterize the closed sets E in the unit circle T which have the property that, for some nondecreasing h: (0, \u221e) \u2192(0, \u221e) with h(0+) = 0, all the Hausdorff h-measure 0 closed sets F \u2286 E are sets of uniqueness (for trigonometric series). In conjunction with K\u00f6rner's result on the existence of Helson sets of multiplicity, this implies the existence of closed sets of multiplicity (M-sets) within which Hausdorff h-measure 0 implies uniqueness, for some h. This is contrasted with the case of closed sets of strict multiplicity (M_0-sets), where results of Ivashev-Musatov and Kaufman establish the opposite.", "doi": "10.2307/2047049", "issn": "0002-9939", "publisher": "American Mathematical Society", "publication": "Proceedings of the American Mathematical Society", "publication_date": "1989-04", "series_number": "4", "volume": "105", "issue": "4", "pages": "894-897" }, { "id": "authors:wj2dt-qcf58", "collection": "authors", "collection_id": "wj2dt-qcf58", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-131009335", "type": "article", "title": "Ordinal Rankings on Measures Annihilating Thin Sets", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Lyons", "given_name": "Russell", "clpid": "Lyons-R" } ], "abstract": "We assign a countable ordinal number to each probability measure which annihilates all H-sets. The descriptive-set theoretic structure of this assignment allows us to show that this class of measures is coanalytic non-Borel. In addition, it allows us to quantify the failure of Rajchman's conjecture. Similar results are obtained for measures annihilating Dirichlet sets.", "doi": "10.2307/2000989", "issn": "0002-9947", "publisher": "American Mathematical Society", "publication": "Transactions of the American Mathematical Society", "publication_date": "1988-12", "series_number": "2", "volume": "310", "issue": "2", "pages": "747-758" }, { "id": "authors:cbnmb-yzm17", "collection": "authors", "collection_id": "cbnmb-yzm17", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-105652249", "type": "book_section", "title": "The Descriptive Set Theory of \u03c3-Ideals of Compact Sets", "book_title": "Logic Colloquium 1988: Proceedings of the Colloquium", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Ferro", "given_name": "R.", "clpid": "Ferro-R" }, { "family_name": "Bonotto", "given_name": "C.", "clpid": "Bonotto-C" }, { "family_name": "Valentini", "given_name": "S.", "clpid": "Valentini-S" }, { "family_name": "Zanardo", "given_name": "S.", "clpid": "Zanardo-A" } ], "abstract": "We present here a survey of the definability theory of \u03c3-ideals of closed sets in compact metric spaces and its structural implications. We also include proofs of\nsome recent results in this area.", "isbn": "978-0-444-87455-9", "publisher": "Elsevier Science", "place_of_publication": "Amsterdam", "publication_date": "1988-08", "pages": "117-138" }, { "id": "authors:kj462-ey512", "collection": "authors", "collection_id": "kj462-ey512", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130612-072000863", "type": "book_section", "title": "\"AD + uniformization\" is equivalent to \"half ad_R\"", "book_title": "Cabal Seminar 81-85", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Martin", "given_name": "Donald A.", "clpid": "Martin-D-A" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "abstract": "Let w be the set of natural numbers and R = 2^\u03c9 the set of \"reals\" . As usual AD is the assertion that every game on \u03c9 is determined and AD_R is the assertion that every game on R\nis determined. Let us also consider an intermediate principle AD^(1/2)_R, which is the assertion that every game in which one of the players plays in \u03c9 and the other in R is determined. Such games appear often in applications instead of full games on R, see for instance [K].", "doi": "10.1007/BFb0084971", "isbn": "978-3-540-50020-9", "publisher": "Springer", "place_of_publication": "Berlin", "publication_date": "1988", "pages": "98-102" }, { "id": "authors:bbn7y-77b26", "collection": "authors", "collection_id": "bbn7y-77b26", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130612-074122964", "type": "book_section", "title": "A coding theorem for measures", "book_title": "Cabal Seminar 81\u201385", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Martin", "given_name": "Donald A.", "clpid": "Martin-D-A" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "abstract": "Assuming ZF + DC + AD Moschovakis (see [Ml]) has shown that if there is a surjection \u03c0 : R \u2192 \u03bb from the reals (R = \u03c9^\u03c9 in this paper) onto an ordinal \u03bb, then\nthere is a surjection \u03c0^* : R \u2192 p(\u03bb) from the reals onto the power set of \u03bb. Let us denote by \u03b2(\u03bb) the set of ultrafilters on \u03bb. The question was raised whether there is an analog of Moschovakis' Theorem for \u03b2(\u03bb), i.e. if there is a surjection from R onto \u03bb, is there one from R onto \u03b2(\u03bb)? Martin showed that this cannot be proved in ZF + DC + AD alone because if V = L(R) and \u03bb= ol, there is no surjection of R onto \u03b2(\u03bb).", "doi": "10.1007/BFb0084972", "isbn": "978-3-540-50020-9", "publisher": "Springer", "place_of_publication": "Berlin", "publication_date": "1988", "pages": "103-109" }, { "id": "authors:ydkxr-ean17", "collection": "authors", "collection_id": "ydkxr-ean17", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130612-090419888", "type": "book_section", "title": "Subsets of \u2135_1 constructible from a real", "book_title": "Cabal Seminar 81-85", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Martin", "given_name": "Donald A.", "clpid": "Martin-D-A" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "abstract": "The purpose of this paper is to give a necessary and sufficient condition for a subset of \u2135_1 to be constructible from a real in terms of structural properties of the code set of A, valid under the condition that an appropriate measurable cardinal exists. This can be combined with recent results of Woodin to provide upper bounds for the consistency strength, of theories of the form\nZFC + \u2200x \u0404 \u03c9^\u03c9(x^# exists)+ \"every subset of \u2135_1 with code set in \u0413 is constructible from a real,\" for various pointclasses \u0413.", "doi": "10.1007/BFb0084973", "isbn": "978-3-540-50020-9", "publisher": "Springer", "place_of_publication": "Berlin", "publication_date": "1988", "pages": "110-116" }, { "id": "authors:hpg12-cvn82", "collection": "authors", "collection_id": "hpg12-cvn82", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-115611198", "type": "article", "title": "The Set of Continuous Functions with the Everywhere Convergent Fourier Series", "author": [ { "family_name": "Ajtai", "given_name": "M.", "clpid": "Ajtai-M" }, { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "This paper deals with the descriptive set theoretic properties of the class EC of continuous functions with everywhere convergent Fourier series. It is shown that this set is a complete coanalytic set in C(T). A natural coanalytic rank function on EC is studied that assigns to each \u0192 \u0404 EC a countable ordinal number, which measures the \"complexity\" of the convergence of the Fourier series of \u0192. It is shown that there exist functions in EC (in fact even differentiable ones) which have arbitrarily large countable rank, so that this provides a proper hierarchy on EC with \u03c9_1 distinct levels.", "doi": "10.2307/2000906", "issn": "0002-9947", "publisher": "American Mathematical Society", "publication": "Transactions of the American Mathematical Society", "publication_date": "1987-07", "series_number": "1", "volume": "302", "issue": "1", "pages": "207-221" }, { "id": "authors:ymnan-6h009", "collection": "authors", "collection_id": "ymnan-6h009", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-103643006", "type": "article", "title": "The Structure of \u03c3-Ideals of Compact Sets", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "A.", "clpid": "Louveau-A" }, { "family_name": "Woodin", "given_name": "W. H.", "clpid": "Woodin-W-H" } ], "abstract": "Motivated by problems in certain areas of analysis, like measure theory and harmonic analysis, where \u03c3-ideals of compact sets are encountered very often as notions of small or exceptional sets, we undertake in this paper a descriptive set theoretic study of \u03c3-ideals of compact sets in compact metrizable spaces. In the first part we study the complexity of such ideals, showing that the structural condition of being a \u03c3-ideal imposes severe definability\nrestrictions. A typical instance is the dichotomy theorem, which states that \u03c3-ideals which are analytic or coanalytic must be actually either complete coanalytic or else G_\u03b4. In the second part we discuss (generators or as we call\nthem here) bases for \u03c3-ideals and in particular the problem of existence of Borel bases for coanalytic non-Borel \u03c3-ideals. We derive here a criterion for the nonexistence of such bases which has several applications. Finally in the\nthird part we develop the connections of the definability properties of \u03c3-ideals with other structural properties, like the countable chain condition, etc.", "doi": "10.2307/2000338", "issn": "0002-9947", "publisher": "American Mathematical Society", "publication": "Transactions of the American Mathematical Society", "publication_date": "1987-05", "series_number": "1", "volume": "301", "issue": "1", "pages": "263-288" }, { "id": "authors:zmh2d-r4j45", "collection": "authors", "collection_id": "zmh2d-r4j45", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130611-134802777", "type": "book_section", "title": "The Complexity of Antidifferentiation, Denjoy Totalization, and Hyperarithmetic Reals", "book_title": "Proceedings of the International Congress of Mathematicians", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Gleason", "given_name": "Andrew M.", "clpid": "Gleason-A-M" } ], "abstract": "We consider real functions on the interval [0, 1]. Denote by \u0394 the set of derivatives; i.e., \u0394 = {\u0192:\u0192 is a derivative}\n= {\u0192 : \u2203F: [0,1]\u2192 R {F is differentiable and \u0192 = F')}.\nIf \u0192 \u0454 \u0394, any F with F' = \u0192 is a primitive of \u0192 and is uniquely determined up to a constant. To normalize, we denote by F(x) = \u0283^x)0 \u0192 the unique primitive of \u0192 with F(0) = 0. This is the original Newtonian concept of integration as the inverse operation of differentiation, i.e., antidifferentiation.", "isbn": "978-0-8218-0110-9", "publisher": "American Mathematical Society", "place_of_publication": "Providence, RI", "publication_date": "1987", "pages": "307-313" }, { "id": "authors:sej0z-a1476", "collection": "authors", "collection_id": "sej0z-a1476", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180815-091126114", "type": "book_section", "title": "Decomposing U-Sets into Simpler Sets", "book_title": "Descriptive Set Theory and the Structure of Sets of Uniqueness", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "abstract": "[no abstract]", "doi": "10.1017/CBO9780511758850.009", "isbn": "9780521358118", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "1987", "pages": "193-230" }, { "id": "authors:6gk44-p3162", "collection": "authors", "collection_id": "6gk44-p3162", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104144223", "type": "book_section", "title": "Sets of resolution and synthesis", "book_title": "Descriptive Set Theory and the Structure of Sets of Uniqueness", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9780511758850.013", "isbn": "9780511758850", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "1987", "pages": "328-348" }, { "id": "authors:7fr4h-wj069", "collection": "authors", "collection_id": "7fr4h-wj069", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104144307", "type": "book_section", "title": "The shrinking method, the theorem of K\u00f6rner and Kaufman, and the solution to the Borel basis problem for U", "book_title": "Descriptive Set Theory and the Structure of Sets of Uniqueness", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9780511758850.010", "isbn": "9780511758850", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "1987", "pages": "231-265" }, { "id": "authors:vxfkq-ezn09", "collection": "authors", "collection_id": "vxfkq-ezn09", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-150902218", "type": "book_section", "title": "Characterizing Rajchman Measures", "book_title": "Descriptive Set Theory and the Structure of Sets of Uniqueness", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "abstract": "[no abstract]", "doi": "10.1017/CBO9780511758850.012", "isbn": "9780521358118", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "1987", "pages": "309-327" }, { "id": "authors:nenak-zhc89", "collection": "authors", "collection_id": "nenak-zhc89", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104144661", "type": "book_section", "title": "Trigonometric series and sets of uniqueness", "book_title": "Descriptive Set Theory and the Structure of Sets of Uniqueness", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9780511758850.004", "isbn": "9780511758850", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "1987", "pages": "21-50" }, { "id": "authors:9v9f6-azn58", "collection": "authors", "collection_id": "9v9f6-azn58", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-081409898", "type": "book", "title": "Descriptive Set Theory and the Structure of Sets of Uniqueness", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain" } ], "abstract": "The study of sets of uniqueness for trigonometric series has a long history, originating in the work of Riemann, Heine, and Cantor in the mid-nineteenth century. Since then it has been a fertile ground for numerous investigations involving real analysis, classical and abstract harmonic analysis, measure theory, functional analysis and number theory. In this book are developed the intriguing and surprising connections that the subject has with descriptive set theory. These have only been discovered recently and the authors present here this novel theory which leads to many new results concerning the structure of sets of uniqueness and include solutions to some of the classical problems in this area. In order to make the material accessible to logicians, set theorists and analysts, the authors have covered in some detail large parts of the classical and modern theory of sets of uniqueness as well as the relevant parts of descriptive set theory. Thus the book is essentially self-contained and will make an excellent introduction to the subject for graduate students and research workers in set theory and analysis.", "doi": "10.1017/CBO9780511758850", "isbn": "9780521358118", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "1987" }, { "id": "authors:frge6-mhe67", "collection": "authors", "collection_id": "frge6-mhe67", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180815-100650741", "type": "book_section", "title": "Extended Uniqueness Sets", "book_title": "Descriptive Set Theory and the Structure of Sets of Uniqueness", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "abstract": "[no abstract]", "doi": "10.1017/CBO9780511758850.011", "isbn": "9780521358118", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "1987", "pages": "266-308" }, { "id": "authors:dxph5-twm07", "collection": "authors", "collection_id": "dxph5-twm07", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-104404521", "type": "book_section", "title": "Classification of the Complexity of U", "book_title": "Descriptive Set Theory and the Structure of Sets of Uniqueness", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "abstract": "[no abstract]", "doi": "10.1017/CBO9780511758850.007", "isbn": "9780521358118", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "1987", "pages": "104-138" }, { "id": "authors:ygvzf-qam22", "collection": "authors", "collection_id": "ygvzf-qam22", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104144571", "type": "book_section", "title": "Symmetric perfect sets and the Salem-Zygmund theorem", "book_title": "Descriptive Set Theory and the Structure of Sets of Uniqueness", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9780511758850.006", "isbn": "9780511758850", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "1987", "pages": "80-103" }, { "id": "authors:ygy5y-rxg23", "collection": "authors", "collection_id": "ygy5y-rxg23", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180816-160145462", "type": "book_section", "title": "Introduction", "book_title": "Descriptive Set Theory and the Structure of Sets of Uniqueness", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9780511758850.002", "isbn": "9780511758850", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "1987", "pages": "1-16" }, { "id": "authors:sqt4p-3bw38", "collection": "authors", "collection_id": "sqt4p-3bw38", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104143805", "type": "book_section", "title": "The Piatetski-Shapiro hierarchy of U-sets", "book_title": "Descriptive Set Theory and the Structure of Sets of Uniqueness", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "abstract": "[no abstract]", "doi": "10.1017/cbo9780511758850.008", "isbn": "9780511758850", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "1987", "pages": "139-192" }, { "id": "authors:b8fmw-7d663", "collection": "authors", "collection_id": "b8fmw-7d663", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180817-104144144", "type": "book_section", "title": "Preface", "book_title": "Descriptive Set Theory and the Structure of Sets of Uniqueness", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "abstract": "This book grew out of a set of notes prepared during the course of a joint Caltech-UCLA Seminar in Descriptive Set Theory and Harmonic Analysis, organized by the authors during the academic year 1985\u201386. We appreciate very much the help as well as the patience of the participants in this seminar. \n\nWe are grateful to G. Debs, R. Dougherty, S. Jackson, R. Kaufman, R. Lyons, and J. Saint Raymond for many valuable comments and suggestions. The first author is indebted to S. Pichorides for introducing him to the subject of uniqueness for trigonometric series. We would like also to thank N. O'Connor for her efficiency, care and patience in typing the manuscript. \n\nThe work of A. S. Kechris has been partially supported by NSF Gratnt DMS84-16349. A. Louveau has been supported by CNRS, France and by UCLA during his visit in the academic year 1985\u201386. He takes this opportunity to thank the Mathematics Department for its hospitality.", "doi": "10.1017/cbo9780511758850.001", "isbn": "9780511758850", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "1987", "pages": "vii" }, { "id": "authors:bhcwx-3xn71", "collection": "authors", "collection_id": "bhcwx-3xn71", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20180810-084749723", "type": "book_section", "title": "The Algebra A of Functions with Absolutely Convergent Fourier Series, Pseudofunctions and Pseudomeasures", "book_title": "Descriptive Set Theory and the Structure of Sets of Uniqueness", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Louveau", "given_name": "Alain", "clpid": "Louveau-A" } ], "abstract": "[no abstract]", "doi": "10.1017/CBO9780511758850.005", "isbn": "9780521358118", "publisher": "Cambridge University Press", "place_of_publication": "Cambridge", "publication_date": "1987", "pages": "51-79" }, { "id": "authors:xaers-p8324", "collection": "authors", "collection_id": "xaers-p8324", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-145231519", "type": "article", "title": "Ranks of differentiable functions", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Woodin", "given_name": "W. Hugh", "clpid": "Woodin-W-H" } ], "abstract": "The purpose of this paper is to define and study a natural rank function\nwhich associates to each differentiable function (say on the interval [0, 1]) a\ncountable ordinal number, which measures the complexity of its derivative.\nFunctions with continuous derivatives have the smallest possible rank 1, a\nfunction like x^2 sin (x^(-1)) has rank 2, etc., and we show that functions of any\ngiven countable ordinal rank exist. This exhibits an underlying hierarchical\nstructure of the class of differentiable functions, consisting of \u03c9_1 distinct levels.\nThe definition of rank is invariant under addition of constants, and so it\nnaturally assigns also to every derivative a unique rank, and an associated\nhierarchy for the class of all derivatives.\n\nThe set D of functions in C[0, 1] which are everywhere differentiable is a\ncomplete coanalytic (and thus non-Borel) set (Mazurkiewicz [Maz]; see Section\n2 below) and it will tum out that the rank function we define has the\nright descriptive set theoretic properties summarized in the concept of a\ncoanalytic norm, explained in Section 1.\n\nOur original description of the rank function was in terms of wellfounded\ntrees and is given in Section 4. In Section 3 we give an equivalent description\nin terms of a Cantor-Bendixson type analysis. We would like to acknowledge\nhere the contribution of D. Preiss. It was in a conversation with one of the\nauthors that this equivalent description was formulated.", "doi": "10.1112/S0025579300011244", "issn": "0025-5793", "publisher": "University College London", "publication": "Mathematika", "publication_date": "1986-12", "series_number": "2", "volume": "33", "issue": "2", "pages": "252-278" }, { "id": "authors:k35mp-zkr91", "collection": "authors", "collection_id": "k35mp-zkr91", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-101108599", "type": "article", "title": "On the Relative Consistency Strength of Determinacy Hypothesis", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Solovay", "given_name": "Robert M.", "clpid": "Solovay-R-M" } ], "abstract": "For any collection of sets of reals C, let C-DET be the statement that all sets of reals in C are determined. In this paper we study questions of the form: For given C \u2286 C', when is C'-DET equivalent, equiconsistent or strictly stronger in consistency strength than C-DET (modulo ZFC)? We focus especially on classes C contained in the projective sets.", "doi": "10.2307/1999790", "issn": "0002-9947", "publisher": "American Mathematical Society", "publication": "Transactions of the American Mathematical Society", "publication_date": "1985-07", "series_number": "1", "volume": "290", "issue": "1", "pages": "179-211" }, { "id": "authors:zx0q4-q1d11", "collection": "authors", "collection_id": "zx0q4-q1d11", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-152728059", "type": "article", "title": "Determinacy with Complicated Strategies", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "For any class of functions F from R into R, AD(F) is the assertion that in every two person game on integers one of the two players has a winning strategy in the class F. It is shown, in ZF + DC + V = L(R), that for any F of cardinality < 2^(N0)(i.e. any F which is a surjective image of R) AD(F) implies AD (the Axiom of Determinacy).", "doi": "10.1090/S0002-9939-1985-0784188-0", "issn": "0002-9939", "publisher": "American Mathematical Society", "publication": "Proceedings of the American Mathematical Society", "publication_date": "1985-06", "series_number": "2", "volume": "94", "issue": "2", "pages": "333-336" }, { "id": "authors:g4q2z-jzn28", "collection": "authors", "collection_id": "g4q2z-jzn28", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130611-130509199", "type": "book_section", "title": "Determinacy and the Structure of L(R)", "book_title": "Recursion Theory", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Nerode", "given_name": "Anil", "clpid": "Nerode-A" }, { "family_name": "Shore", "given_name": "Richard A.", "clpid": "Shore-R-A" } ], "abstract": "Let \u03c9 = {0, 1, 2, ... } be the set of natural numbers and R = \u03c9^\u03c9 the set of all infinite sequences from \u03c9, or for simplicity reals. To each set A \u2286 R we associate a two-person infinite game, in which players I and II alternatively play natural numbers I x(0) x(2) II x(1) x(3)...x(O), x(l), x(2), ... and if x is the real they eventually produce, then I wins iff x \u0454 A. The notion of a winning strategy for player I or II is defined in the usual\nway, and we call A determined if either player I or player II has a winning strategy in the above game. For a collection \u2308 of sets of reals let \u2308-DET be the statement\nthat all sets A \u0454 \u2308 are determined. Finally AD (The Axiom of Determinacy) is the\nstatement that all sets of reals are determined.", "doi": "10.1090/pspum/042", "isbn": "978-0-8218-1447-5", "publisher": "American Mathematical Society", "place_of_publication": "Providence, RI", "publication_date": "1985", "pages": "271-283" }, { "id": "authors:c2w35-3pm59", "collection": "authors", "collection_id": "c2w35-3pm59", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-102414726", "type": "book_section", "title": "Sets of everywhere singular functions", "book_title": "Recursion theory week", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Ebbinghaus", "given_name": "Heinz-Dieter", "clpid": "Ebbinghaus-H-D" }, { "family_name": "M\u00fcller", "given_name": "G. H.", "clpid": "M\u00fcller-G-H" }, { "family_name": "Sacks", "given_name": "Gerald E.", "clpid": "Sacks-G-E" } ], "abstract": "In this paper we present a simple general\nmethod for demonstrating that in certain function spaces various sets\nconsisting of functions that exhibit at every point a prescribed kind\nof singularity form a coanalytic but not Borel set. We illustrate this\nmethod by providing new proofs that the set of nowhere differential\ncontinuous functions on [0,1] is (coanalytic but) not Borel and similarly\nfor the set of continuous functions on [0,1] which fail everywhere\nto have a unilateral derivative (including \u00b1 \u221e) -- the so called\nBesicovitch functions. These results were originally proved by Mauldin,\n[Mau] and unpublished, respectively. We also give a new example of a\ncoanalytic not Borel set, namely the set of integrable functions with\neverywhere divergent Fourier series. Finally, we formulate an abstract\ntheorem, which includes as simple instances all the above and other\nsimilar examples.", "doi": "10.1007/BFb0076223", "isbn": "978-3-540-15673-4", "publisher": "Springer-Verlag", "place_of_publication": "Berlin", "publication_date": "1985", "pages": "233-244" }, { "id": "authors:afzqs-0pg39", "collection": "authors", "collection_id": "afzqs-0pg39", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-083650112", "type": "article", "title": "The Axiom of Determinacy Implies Dependent Choices in L(R)", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We prove the following Main Theorem: ZF+AD+V=L(R)\u21d2DC. As a corollary we have that Con(ZF+AD)\u21d2Con(ZF+AD+DC). Combined with the result of Woodin that Con(ZF+AD)\u21d2Con(ZF+AD+\u00acAC^\u03c9) it follows that DC (as well as AC^\u03c9) is independent relative to ZF+AD. It is finally shown (jointly with H. Woodin) that ZF+AD+\u00acDC_R, where DC_R is DC restricted to reals, implies the consistency of ZF+AD+DC, in fact implies R^# (i.e. the sharp of L(R)) exists.", "issn": "0022-4812", "publisher": "Association for Symbolic Logic", "publication": "Journal of Symbolic Logic", "publication_date": "1984-03", "series_number": "1", "volume": "49", "issue": "1", "pages": "161-173" }, { "id": "authors:x2f6s-mqc03", "collection": "authors", "collection_id": "x2f6s-mqc03", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130611-104220186", "type": "book_section", "title": "Sets of ordinals constructible from trees and the third Victoria Delfino problem", "book_title": "Axiomatic Set Theory", "author": [ { "family_name": "Becker", "given_name": "Howard S.", "clpid": "Becker-H-S" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Baumgartner", "given_name": "James E.", "clpid": "Baumgartner-J-E" }, { "family_name": "Martin", "given_name": "Donald A.", "clpid": "Martin-D-A" }, { "family_name": "Shelah", "given_name": "Saharon", "clpid": "Shelah-S" } ], "abstract": "A very important part of the structure theory of \u03a3^1_2 sets of reals is based on their close interrelationship with the G\u00f6del constructible universe L. The fundamental fact underlying this connection is the theorem of Shoenfield\nwhich asserts that every \u03a3^1_2 set of reals is Souslin over L. This means that given any \u03a3^1_2 subset of the reals (=\u03c9^\u03c9 in this paper), there is a tree T on \u03c9 x \u03bb (\u03bb some ordinal, which can be taken to be \u2135_l here) such that T \u0454 L and A = p[T] = {\u0251 \u0454 \u03c9^\u03c9: : \u2203f \u0454 \u03bb^\u03c9 \u2200n(\u0251\u21ben,f\u21ben) \u0454 T}.", "doi": "10.1090/conm/031", "isbn": "978-0-8218-5026-8", "publisher": "American Mathematical Society", "place_of_publication": "Providence, RI", "publication_date": "1984", "pages": "13-29" }, { "id": "authors:6x4k8-ghn42", "collection": "authors", "collection_id": "6x4k8-ghn42", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:KECpnas83", "type": "article", "title": "Equivalence of partition properties and determinacy", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Woodin", "given_name": "W. Hugh", "clpid": "Woodin-W-H" } ], "abstract": "It is shown that, within L(R), the smallest inner model of set theory containing the reals, the axiom of determinacy is equivalent to the existence of arbitrarily large cardinals below \u0398 with the strong partition property \u0138 \u2192 (\u0138)^\u0138.", "pmcid": "PMC393690", "issn": "0027-8424", "publisher": "National Academy of Sciences", "publication": "Proceedings of the National Academy of Sciences of the United States of America", "publication_date": "1983-03-15", "series_number": "6", "volume": "80", "issue": "6", "pages": "1783-1786" }, { "id": "authors:gnmzt-mg363", "collection": "authors", "collection_id": "gnmzt-mg363", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130531-151329535", "type": "book_section", "title": "Introduction to Q-Theory", "book_title": "Cabal Seminar 79-81", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Martin", "given_name": "Donald A.", "clpid": "Martin-D-A" }, { "family_name": "Solovay", "given_name": "Robert M.", "clpid": "Solovay-R-M" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Martin", "given_name": "D. A.", "clpid": "Martin-D-A" }, { "family_name": "Moschovakis", "given_name": "Yiannis N.", "clpid": "Moschovakis-Y-N" } ], "abstract": "Working in the context of Projective Determinacy (PD), we introduce and study in this paper a countable \u220f^1_(2n+1) set of reals Q_(2n+l) and an associated real y^0_(2n+l) for each n \u2265 0 (real means element of \u03c9^\u03c9 in this paper). Our theory has analytical (descriptive set theoretic) as well as set theoretic aspects, strongly interrelated with each other.", "doi": "10.1007/BFb0071702", "isbn": "978-3-540-12688-1", "publisher": "Springer", "place_of_publication": "Berlin", "publication_date": "1983", "pages": "199-282" }, { "id": "authors:kbpn3-d4360", "collection": "authors", "collection_id": "kbpn3-d4360", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130521-101019031", "type": "book_section", "title": "Ordinal games and their applications", "book_title": "Logic, methodology, and philosophy of science VI", "author": [ { "family_name": "Harrington", "given_name": "Leo A.", "clpid": "Harrington-L-A" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Cohen", "given_name": "L. Jonathan", "clpid": "Cohen-L-J" } ], "abstract": "In this paper we give a survey of certain results on the determinacy of\ngames on ordinals and its various applications. A more comprehensive\ntreatment including more detailed versions, extensions and proofs of the\ntheorems mentioned here, will appear in a forthcoming paper by the\nauthors.", "isbn": "0-444-85423-1", "publisher": "North-Holland Publishing Co.", "place_of_publication": "Amsterdam", "publication_date": "1982", "pages": "273-277" }, { "id": "authors:v7eg0-07r27", "collection": "authors", "collection_id": "v7eg0-07r27", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130531-145645449", "type": "book_section", "title": "Effective Ramsey Theorems in the Projective Hierarchy", "book_title": "Proceedings of the Herbrand Symposium", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Stern", "given_name": "Jacques", "clpid": "Stern-J" }, { "family_name": "Herbrand", "given_name": "Jacques", "clpid": "Herbrand-J" } ], "abstract": "This chapter discusses Ramsey theorems that are effective in the projective hierarchy. The application of Q-theory is quite elementary; it will bring out some of its essential ideas and methods that can be basic ingredients in more elaborate uses and applications of this theory. The chapter proves the main theorem and discusses results related results on \u0394^1_(2n+l)- and Q_(2n+l)- encodability, and discusses some related open problems. One of the proof uses Mathias forcing over an appropriate inner model of ZFC.", "doi": "10.1016/S0049-237X(08)71883-4", "isbn": "9780444864178", "publisher": "Elsevier", "place_of_publication": "Amsterdam", "publication_date": "1982", "pages": "179-187" }, { "id": "authors:1ttt9-hw631", "collection": "authors", "collection_id": "1ttt9-hw631", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-105719531", "type": "article", "title": "Forcing with \u0394 perfect trees and minimal \u0394-degrees", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "This paper is a sequel to [3] and it contains, among other things, proofs of the results announced in the last section of that paper. In \u00a71, we use the general method of [3] together with reflection arguments to study the properties of forcing with \u0394 perfect trees, for certain Spector pointclasses \u0393, obtaining as a main result the existence of a continuum of minimal \u0394-degrees for such \u0393's, under determinacy hypotheses. In particular, using PD, we prove the existence of continuum many minimal \u0394^(1)_(2n+1)-degrees, for all n.^(2) Following an idea of Leo Harrington, we extend these results in \u00a72 to show the existence of minimal strict upper bounds for sequences of \u0394-degrees which are not too far apart. As a corollary, it is computed that the length of the natural hierarchy of \u0394^(1)_(2n+1)-degrees is equal to \u03c9 when n \u2265 1. (By results of Sacks and Richter the length of the natural hierarchy of \u0394^(1)_(1)-degrees is known to be equal to the first recursively inaccessible ordinal.)", "issn": "0022-4812", "publisher": "Association for Symbolic Logic", "publication": "Journal of Symbolic Logic", "publication_date": "1981-12", "series_number": "4", "volume": "46", "issue": "4", "pages": "803-816" }, { "id": "authors:m56ts-pbx63", "collection": "authors", "collection_id": "m56ts-pbx63", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-085823503", "type": "article", "title": "On the determinacy of games on ordinals", "author": [ { "family_name": "Harrington", "given_name": "Leo A.", "clpid": "Harrington-L-A" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "Let \u03bb be an ordinal and A \u2286 \u03bb^\u03c9 x \u03bb^\u03c9. As usual we associate with it the game G(A;\u03bb): I II I and II alternatively \u03be0 \u03b70 play \u03be0, \u03b70, \u03be1,\u03b71,.... from \u03bb; \u03be1 I wins iff (\u03be, \u03ae) \u03b5A.\n\u03b71 \u03be \u03ae.", "doi": "10.1016/0003-4843(81)90001-2", "issn": "0003-4843", "publisher": "Elsevier", "publication": "Annals of Mathematical Logic", "publication_date": "1981-06", "series_number": "2", "volume": "20", "issue": "2", "pages": "109-154" }, { "id": "authors:0sc9r-yb091", "collection": "authors", "collection_id": "0sc9r-yb091", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130531-144033005", "type": "book_section", "title": "Souslin cardinals, \u03ba-souslin sets and the scale property in the hyperprojective hierarchy", "book_title": "Cabal Seminar 77-79", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Martin", "given_name": "Donald A.", "clpid": "Martin-D-A" }, { "family_name": "Moschovakis", "given_name": "Yiannis N.", "clpid": "Moschovakis-Y-N" } ], "abstract": "Let \u0398 = sup{\u03be : \u03be is the length of a prewellordering of the set of reals R(= \u03c9^\u03c9)}. Let K < \u0398 be an infinite cardinal. The class \u00a7(k) of k-Souslin sets has some well-known closure properties, i.e. it is closed under continuous substitutions , countable intersections and unions, and existential quantification over R.", "doi": "10.1007/BFb0090238", "isbn": "978-3-540-10288-5", "publisher": "Springer-Verlag", "place_of_publication": "Berlin", "publication_date": "1981", "pages": "127-146" }, { "id": "authors:hayep-6ny35", "collection": "authors", "collection_id": "hayep-6ny35", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130531-114321828", "type": "book_section", "title": "The axiom of determinacy, strong partition properties and nonsingular measures", "book_title": "Cabal Seminar 77-79", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Kleinberg", "given_name": "Eugene M.", "clpid": "Kleinberg-E-M" }, { "family_name": "Moschovakis", "given_name": "Yiannis N.", "clpid": "Moschovakis-Y-N" }, { "family_name": "Woodin", "given_name": "W. Hugh", "clpid": "Woodin-W-H" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Kleinberg", "given_name": "Eugene M.", "clpid": "Kleinberg-E-M" }, { "family_name": "Moschovakis", "given_name": "Yiannis N.", "clpid": "Moschovakis-Y-N" } ], "abstract": "In this paper we study the relationship between AD and strong partition properties of cardinals as well as some consequences of these properties themselves.", "doi": "10.1007/BFb0090236", "isbn": "978-3-540-10288-5", "publisher": "Springer-Verlag", "place_of_publication": "Berlin", "publication_date": "1981", "pages": "75-99" }, { "id": "authors:4ftxr-8eb09", "collection": "authors", "collection_id": "4ftxr-8eb09", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130531-131451792", "type": "book_section", "title": "The axiom of determinacy and the prewellordering property", "book_title": "Cabal Seminar 77-79", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Solovay", "given_name": "Robert M.", "clpid": "Solovay-R-M" }, { "family_name": "Steel", "given_name": "John R.", "clpid": "Steel-J-R" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Martin", "given_name": "Donald A.", "clpid": "Martin-D-A" }, { "family_name": "Moschovakis", "given_name": "Yiannis N.", "clpid": "Moschovakis-Y-N" } ], "abstract": "Let \u03c9 = (0,1,2,...) be the set of natural numbers and R = \u03c9^\u03c9 the set of all functions from \u03c9 into \u03c9, or for simplicity reals. A product space is of the form\nX = X_1 x X_2 x ... x X_k, where X_1 = \u03c9 or R. Subsets of these product spaces are called pointsets. A boldface pointclass is a class of pointsets closed under continuous preimages and containing all clopen pointsets (in all product spaces).", "doi": "10.1007/BFb0090237", "isbn": "978-3-540-10288-5", "publisher": "Springer-Verlag", "place_of_publication": "Berlin", "publication_date": "1981", "pages": "101-125" }, { "id": "authors:y8ge7-nvp84", "collection": "authors", "collection_id": "y8ge7-nvp84", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130531-110245492", "type": "book_section", "title": "Homogeneous trees and projective scales", "book_title": "Cabal Seminar 77-79", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Martin", "given_name": "Donald A.", "clpid": "Martin-D-A" }, { "family_name": "Moschovakis", "given_name": "Yiannis N.", "clpid": "Moschovakis-Y-N" } ], "abstract": "This exposition is a sequel to Kechris [1978]. Its main purpose is to show how set theoretical techniques, among them infinite exponent partition relations can be used to produce homogeneous trees for projective sets. The work here is again understood as being carried completely within L[R], with the hypothesis that AD+DC holds in this model.", "doi": "10.1007/BFb0090235", "isbn": "978-3-540-10288-5", "publisher": "Springer-Verlag", "place_of_publication": "Berlin", "publication_date": "1981", "pages": "33-73" }, { "id": "authors:y5yry-bxe69", "collection": "authors", "collection_id": "y5yry-bxe69", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130529-162228399", "type": "book_section", "title": "A note on Wadge degrees", "book_title": "Cabal Seminar 77-79", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Kechris", "given_name": "A. S.", "clpid": "Kechris-A-S" }, { "family_name": "Martin", "given_name": "Donald A.", "clpid": "Martin-D-A" }, { "family_name": "Moschovakis", "given_name": "Yiannis N.", "clpid": "Moschovakis-Y-N" } ], "abstract": "For pertinent definitions see published works of the Cabal Seminar. It has been shown by D. A. Martin that under the assumption of AD, Wadge degrees are well ordered; for Wadge degrees of \u0394^(1)_(2n) sets \u0394^(1)_(2n)-determinacy suffices. Martin has also proved (using AD) that the length of the well ordering of \u0394^(1)_(2n) Wadge degrees is exactly \u03b4^(1)_(2n+1). The author gives an alternate proof of this fact, using only \u0394^(1)_(2n)-determinacy.", "doi": "10.1007/BFb0090240", "isbn": "978-3-540-38422-9", "publisher": "Springer-Verlag", "place_of_publication": "Berlin", "publication_date": "1981", "pages": "165-168" }, { "id": "authors:chgah-wa551", "collection": "authors", "collection_id": "chgah-wa551", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130531-104437708", "type": "book_section", "title": "Recent Advances in the Theory of Higher Level Projective Sets", "book_title": "The Kleene Symposium", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Barwise", "given_name": "J.", "clpid": "Barwise-J" }, { "family_name": "Keisler", "given_name": "H. J.", "clpid": "Keisler-H-J" }, { "family_name": "Kunen", "given_name": "K.", "clpid": "Kunen-K" } ], "abstract": "An outline is given of certain aspects of the current theory of higher level projective sets, especially concentrating on the structure theory of II^1_3 sets, from determinacy hypotheses. Some of the key open problems in this area are also discussed.", "doi": "10.1016/S0049-237X(08)71258-8", "isbn": "978-0-444-85345-5", "publisher": "Elsevier", "place_of_publication": "Amsterdam", "publication_date": "1980", "pages": "149-166" }, { "id": "authors:0ypgb-1hd98", "collection": "authors", "collection_id": "0ypgb-1hd98", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130610-140456846", "type": "article", "title": "An overview of descriptive set theory", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "Understanding the structure of sets of reals is one of the\nfundamental problems of set theory.\nOf course, an arbitrary subset of the continuum is immensely complicated,\nand exhibits in general no regularities. For example it could be non(Lebesgue)-measurable\nor fail to have the property of Baire. So from the early days of\nset theory, about a hundred years ago, mathematicians have concentrated on the\nstudy of various classes of definable sets of reals, wide enough to include the\nusual sets appearing in analysis, topology, etc., but also restricted to possibly\nexhibit a regular behaviour.\nThe study of such classes of definable sets is the purpose of descriptive\nset theory. Indeed, to a large extent descriptive set theory is mainly concerned\nwith the structure of the projective sets of reals, which we proceed to define.", "issn": "1151-1745", "publisher": "Universit\u00e9 Pierre et Marie Curie", "publication": "Publications Math\u00e9matiques de l'Universit\u00e9 Pierre et Marie Curie", "publication_date": "1979-02", "volume": "4" }, { "id": "authors:9ewfk-rwc14", "collection": "authors", "collection_id": "9ewfk-rwc14", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-081838953", "type": "article", "title": "Countable ordinals and the analytical hierarchy, II", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "This paper is a contribution to the study of projective sets under the hypothesis of definable determinacy. For our purposes here this can be understood as the hypothesis every set of rea1s in L[\u03c9^\u03c9] (= the smallest inner model of ZF containing \u03c9^\u03c9) is determined. Since questions about the projective hierarchy are absolute between the real world and\nL[\u03c9^\u03c9] it is innocuous to work entirely within L[\u03c9^\u03c9].", "doi": "10.1016/0003-4843(78)90010-4", "issn": "0003-4843", "publisher": "Elsevier", "publication": "Annals of Mathematical Logic", "publication_date": "1978-12", "series_number": "3", "volume": "15", "issue": "3", "pages": "193-223" }, { "id": "authors:4c62c-cpc60", "collection": "authors", "collection_id": "4c62c-cpc60", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-081912845", "type": "article", "title": "The Perfect Set Theorem and Definable Wellorderings of the Continuum", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "Let \u0393 be a collection of relations on the reals and let M be a set of reals. We call M a perfect set basis for \u0393 if every set in \u0393 with parameters from M which is not totally included in M contains a perfect subset with code in M. A simple elementary proof is given of the following result (assuming mild regularity conditions on \u0393 and M): If M is a perfect set basis for \u0393, the field of every wellordering in \u0393 is contained in M. An immediate corollary is Mansfield's Theorem that the existence of a \u03a3^1_2 wellordering of the reals implies that every real is constructible. Other applications and extensions of the main result are also given.", "issn": "0022-4812", "publisher": "Association for Symbolic Logic", "publication": "Journal of Symbolic Logic", "publication_date": "1978-12", "series_number": "4", "volume": "43", "issue": "4", "pages": "630-634" }, { "id": "authors:rbpcw-3vz95", "collection": "authors", "collection_id": "rbpcw-3vz95", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-083000082", "type": "article", "title": "Minimal Upper Bounds for Sequences of \u0394^1_(2n)-Degrees", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "It is proved here, assuming Projective Determinacy, that every ascending sequence of \u0394^1_(2n)-degrees has a minimal strict upper bound but no least strict upper bound. This generalizes a result of Friedman for n = 1.", "issn": "0022-4812", "publisher": "Association for Symbolic Logic", "publication": "Journal of Symbolic Logic", "publication_date": "1978-09", "series_number": "3", "volume": "43", "issue": "3", "pages": "502-507" }, { "id": "authors:x826v-knz27", "collection": "authors", "collection_id": "x826v-knz27", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130625-140857081", "type": "article", "title": "On the theory of \u220f_3^1 sets of reals", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Martin", "given_name": "D. A.", "clpid": "Martin-D-A" } ], "abstract": "Assuming that \u2200x \u0404 \u03c9^\u03c9 (x^# exists), let u_\u0251 be the \u0251th uniform indiscernible (see [3] or [2] ). A canonical coding system for ordinals < u_\u03c9 can be defined by letting W0_\u03c9 = {w \u0404 \u03c9^\u03c9: w = (n, x^#), for some n \u0404 \u03c9, x \u0404 \u03c9^\u03c9} and for w = (n, x^#) \u0454 W0_\u03c9, \u2502w\u2502 = \u01ac^L_n [x](u_l',... , u_k_n),\nwhere T_n is the nth term in a recursive enumeration of all terms in the language of ZF + V = L [x], x a constant, taking always ordinal values. Call a relation P(\u03be x), where ~varies over u^\u03c9 and x over \u03c9^\u03c9, \u220f^1_k if P^*(w, x)\u21d4 w \u0404 W0_\u03c9 \u039b P(\u2502w\u2502, x) is \u220f^1_k. An ordinal \u03be < u_\u03c9 is called \u0394^1_k if it has a \u0394^1_k notation i.e. \u2203 w \u0404 W0_\u03c9 (w \u0404 \u0394^1_k \u039b \u2502w\u2502 = \u03be).", "doi": "10.1090/S0002-9904-1978-14447-4", "issn": "0273-0979", "publisher": "American Mathematical Society", "publication": "Bulletin of the American Mathematical Society", "publication_date": "1978-01", "series_number": "1", "volume": "84", "issue": "1", "pages": "149-151" }, { "id": "authors:8v9kx-cf045", "collection": "authors", "collection_id": "8v9kx-cf045", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-094742588", "type": "book_section", "title": "Notes on the Theory of Scales", "book_title": "Cabal Seminar 76\u201377", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Moschovakis", "given_name": "Yiannis N.", "clpid": "Moschovakis-Y-N" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Moschovakis", "given_name": "Yiannis N.", "clpid": "Moschovakis-Y-N" } ], "abstract": "It is the purpose of these notes to give an informal exposition of several recent results in Descriptive Set Theory, all centering around the notion of a scale. This was first isolated explicitly in the generalization of the Uniformization Theorem on the hypothesis of projective determinacy [14], but is surely implicit in some of the classical proofs. It has turned out that scales have many applications beyond the Uniformization Theorem, both in producing new results and in providing more elegant proofs of known results.", "doi": "10.1007/BFb0069294", "isbn": "978-3-540-09086-1", "publisher": "Springer", "place_of_publication": "Berlin", "publication_date": "1978", "pages": "1-53" }, { "id": "authors:t1egj-qk440", "collection": "authors", "collection_id": "t1egj-qk440", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130531-074153858", "type": "book_section", "title": "On Spector classes", "book_title": "Cabal Seminar 76\u201377", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Kechris", "given_name": "Alexander S.", "clpid": "Kechris-A-S" }, { "family_name": "Moschovakis", "given_name": "Yiannis N.", "clpid": "Moschovakis-Y-N" } ], "abstract": "This paper contains an exposition of certain aspects of the theory of Spector classes. The general plan is as follows: In Section 1 we present a review of the structure theory, mainly developed in Chapter 9 of Moschovakis [1]. In Section 2 we give a comprehensive list of examples from various areas of definability theory. Finally, in Section 3, we deal with characterization and representation theorems.", "doi": "10.1007/BFb0069303", "isbn": "9783540090861", "publisher": "Springer", "place_of_publication": "Berlin", "publication_date": "1978", "pages": "245-277" }, { "id": "authors:w3qp7-s7w16", "collection": "authors", "collection_id": "w3qp7-s7w16", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-100408397", "type": "book_section", "title": "Spector second order classes and reflection", "book_title": "Generalized recursion theory II", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Fenstad", "given_name": "Jens Erik", "clpid": "Fenstad-J-E" }, { "family_name": "Gandy", "given_name": "R. O.", "clpid": "Gandy-R-O" }, { "family_name": "Sacks", "given_name": "Gerald E.", "clpid": "Sacks-G-E" } ], "abstract": "The purpose of this paper, which grew out of a series of lectures (short course)\ndelivered at the Oslo Conference, is to give a survey of the theory of reflection\nin the context of Spector second order classes. We have mainly aimed at presenting\na global view of a rather extensive part of definability theory and at the\nsame time illustrate how the main concepts and results to be presented below unify\na large number of diverse areas in this theory and also clarify many of its important\naspects.", "isbn": "9780444851635", "publisher": "North-Holland Publishing Company", "place_of_publication": "Amsterdam", "publication_date": "1978", "pages": "147-183" }, { "id": "authors:tfk12-c4354", "collection": "authors", "collection_id": "tfk12-c4354", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130530-070413635", "type": "book_section", "title": "On Transfinite Sequences of Projective Sets with an Application to \u2211^1_2 Equivalence Relations", "book_title": "Logic Colloquium '77", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Macintyre", "given_name": "Angus", "clpid": "Macintyre-A" }, { "family_name": "Pacholski", "given_name": "Leszek", "clpid": "Pacholski-L" }, { "family_name": "Paris", "given_name": "Jeff", "clpid": "Paris-J" } ], "abstract": "This chapter focuses on transfinite sequences of projective sets with an application to \u03a3^1_(\u223c2n) equivalence relations. From the full axiom of determinacy (AD), the chapter demonstrates that the class of \u03a3^1_(\u223c2n) sets is closed under well ordered unions. This result is then applied to compute the maximum length of transfinite 1\u20131 sequences of ^1_(\u223c2n) sets, from AD again. As a further application, the chapter demonstrates that in the real world, every \u03a3^1_2 equivalence relation has either \u2264_\u05d01k, or 2_0^\u05d0 equivalence classes. In addition, the cardinality of the maximum length of a transfinite sequence of ^1_(\u223c2n) sets without repetitions is computed in the chapter.", "doi": "10.1016/S0049-237X(08)71998-0", "isbn": "9780444851789", "publisher": "North Holland Publishing Company", "place_of_publication": "Amsterdam", "publication_date": "1978", "pages": "155-160" }, { "id": "authors:3f73d-4qr63", "collection": "authors", "collection_id": "3f73d-4qr63", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130531-091303529", "type": "book_section", "title": "Forcing in analysis", "book_title": "Higher Set Theory", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "M\u00fcller", "given_name": "Gert H.", "clpid": "M\u00fcller-G-H" }, { "family_name": "Scott", "given_name": "Dana S.", "clpid": "Scott-D-S" } ], "abstract": "In this paper we develop a method for applying \"forcing techniques\" (or equivalently \"category arguments\") in the analytical hierarchy. We present here the basic general theory and some simple applications. In a subsequent publication, which is in preparation, further and technically more complicated applications will be given,\nincluding perfect set forcing and in particular the results in the circulated note [Ke 1], which are summarized in Section 6.", "doi": "10.1007/BFb0103105", "isbn": "978-3-540-08926-1", "publisher": "Springer-Verlag", "place_of_publication": "Berlin", "publication_date": "1978", "pages": "277-302" }, { "id": "authors:bw5br-9vb69", "collection": "authors", "collection_id": "bw5br-9vb69", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-134758796", "type": "book_section", "title": "AD and projective ordinals", "book_title": "Cabal Seminar 76-77", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "contributor": [ { "family_name": "Kechris", "given_name": "A. S.", "clpid": "Kechris-A-S" }, { "family_name": "Moschovakis", "given_name": "Yiannis N.", "clpid": "Moschovakis-Y-N" } ], "abstract": "This is an unpolished exposition of some work in the theory of projective ordinals\nunder the hypothesis of definable determinacy. This is understood here as the\nhypothesis that every set of reals in L[R] is determined. Since the projective\nordinals are absolute between the real world and L[R] we carry this study entirely\nwithin L[R]. Thus we will use the full Axiom of Determinacy (AD) together with\nZF + DC (DC is of course the only choice principle that is preserved under this\ntransition to L[R]).", "doi": "10.1007/BFb0069296", "isbn": "978-3-540-09086-1", "publisher": "Springer", "place_of_publication": "Berlin", "publication_date": "1978", "pages": "91-132" }, { "id": "authors:vymew-27z56", "collection": "authors", "collection_id": "vymew-27z56", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-135105483", "type": "article", "title": "On a notion of smallness for subsets of the Baire space", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "Let us call a set A \u2286 \u03c9^\u03c9 of functions from \u03c9 into \u03c9 \u03c3-bounded if there is a countable sequence of functions (\u03b1_n: n \u0404 \u03c9)\u2286 \u03c9^\u03c9 such that every member of A is pointwise dominated by an element of that sequence. We study in this paper definability questions concerning this notion of smallness for subsets of \u03c9^\u03c9. We show that most of the usual definability results about the structure of countable subsets of \u03c9^\u03c9 have corresponding versions which hold about \u03c3-bounded subsets of \u03c9^\u03c9. For example, we show that every \u03a3_(2n+1^1 \u03c3-bounded subset of \u03c9^\u03c9 has a \u0394_(2n+1)^1 \"bound\" {\u03b1_m: m \u0404 \u03c9} and also that for any n \u2265 0 there are largest \u03c3-bounded \u03a0_(2n+1)^1 and \u03a3_(2n+2)^1 sets. We need here the axiom of projective determinacy if n \u2265 1. In order to study the notion of \u03c3-boundedness a simple game is devised which plays here a role similar to that of the standard ^*-games (see [My]) in the theory of countable sets. In the last part of the paper a class of games is defined which generalizes the ^*- and ^(**)-(or Banach-Mazur) games (see [My]) as well as the game mentioned above. Each of these games defines naturally a notion of smallness for subsets of \u03c9^\u03c9 whose special cases include countability, being of the first category and \u03c3-boundedness and for which one can generalize all the main results of the present paper.", "issn": "0002-9947", "publisher": "American Mathematical Society", "publication": "Transactions of the American Mathematical Society", "publication_date": "1977-05", "volume": "229", "pages": "191-207" }, { "id": "authors:wa8kx-t2j47", "collection": "authors", "collection_id": "wa8kx-t2j47", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-103036161", "type": "article", "title": "Classifying Projective-like Hierarchies", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "The starting point for the investigations reported in this paper is the following surprising result of Steel [St]:\nAssuming AD + DC let \u0413 be a collection of pointsets containing all open sets and closed under continuous preimages and countable intersections and unions. Then either \u0413 or \u0413 ( = the dual of \u0413) has the reduction property.\nThis result raises naturally the question if a similar phenomenon occurs with the stronger prewellordering property, perhaps under stronger closure assumptions on \u0413.", "issn": "0072-7466", "publisher": "Bulletin of the Greek Mathematical Society", "publication": "Bulletin of the Greek Mathematical Society", "publication_date": "1977", "series_number": "2", "volume": "18", "issue": "2", "pages": "254-275" }, { "id": "authors:8hmze-w5k35", "collection": "authors", "collection_id": "8hmze-w5k35", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-111450945", "type": "article", "title": "\u220f^1_2 singletons and O^#", "author": [ { "family_name": "Harrington", "given_name": "Leo", "clpid": "Harrington-L-A" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "A conjecture of Solovay states: Assuming that for every real \u0251, \u0251^# exists, the constructibility degrees of \u220f^1_2 singletons are wellordered and the successor steps in this wellordering are given by the sharps. In this paper we prove among others things that (assuming \u2200\u0251 (\u0251^# exists))\nfor every \u220f^1_2 singleton a either O^# is constructible from \u0251 or \u0251^# is constructible from O^#. From a relativized version of this result it follows that the constructibility degrees of O^#, O^(##), O^(###),... are the first \u03c9 constructibility degrees of sharps of \u220f^1_2 singletons.", "issn": "0016-2736", "publisher": "Institute of Mathematics, Polish Academy of Sciences", "publication": "Fundamenta Mathematicae", "publication_date": "1977", "series_number": "3", "volume": "95", "issue": "3", "pages": "167-171" }, { "id": "authors:836bv-kns72", "collection": "authors", "collection_id": "836bv-kns72", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130624-085131170", "type": "article", "title": "On Monotone vs. Nonmonotone Induction", "author": [ { "family_name": "Harrington", "given_name": "Leo A.", "clpid": "Harrington-L-A" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "Introduction. For definitions and notation in what follows, see [4] and [5].", "issn": "0273-0979", "publisher": "American Mathematical Society", "publication": "Bulletin of the American Mathematical Society", "publication_date": "1976-11", "series_number": "6", "volume": "82", "issue": "6", "pages": "888-890" }, { "id": "authors:225v3-pqd69", "collection": "authors", "collection_id": "225v3-pqd69", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130529-152738529", "type": "article", "title": "A Basis Result for \u03a3^0_3 Sets of Reals with an Application to Minimal Covers", "author": [ { "family_name": "Harrington", "given_name": "Leo A.", "clpid": "Harrington-L-A" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "It is shown that every \u2211^0_3 set of reals which contains reals of arbitrarily high Turing degree in the hyperarithmetic hierarchy contains reals of every Turing degree above the degree of Kleene's O. As an application it is shown that every Turing degree above the Turing degree of Kleene's O is a minimal cover.", "issn": "0002-9939", "publisher": "American Mathematical Society", "publication": "Proceedings of the American Mathematical Society", "publication_date": "1975-12", "series_number": "2", "volume": "53", "issue": "2", "pages": "445-448" }, { "id": "authors:ayz3c-rjf46", "collection": "authors", "collection_id": "ayz3c-rjf46", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130529-073254315", "type": "article", "title": "A note on universal sets for classes of countable G_\u03b4'S", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Martin", "given_name": "D. A.", "clpid": "Martin-D-A" } ], "abstract": "In a recent article [2] D. G. Larman and C. A. Rogers proved the following two results in Descriptive Set Theory (where R = the space of real numbers): (1) There is no analytic set in the plane R^2, which is universal for the countable closed subsets of R; (2) there is no Borel set in R^2, which is universal for the countable G_\u03b4 subsets of R. Recall that, if b is a class of subsets of a space X, a set U \u2286 X \u00d7 X is called universal for C if (\u0251) for each x \u0454 X, U_x = def {y : (x, y) U} \u0454 C, and (b) for each A \u0454 C there is an x such that A = U_x. (Larman and Rogers have also shown that in both cases coanalytic universal sets exist.)", "doi": "10.1112/S0025579300004484", "issn": "0025-5793", "publisher": "University College London", "publication": "Mathematika", "publication_date": "1975-06", "series_number": "1", "volume": "22", "issue": "1", "pages": "43-45" }, { "id": "authors:wvp16-k7081", "collection": "authors", "collection_id": "wvp16-k7081", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130524-135424900", "type": "article", "title": "On Characterizing Spector Classes", "author": [ { "family_name": "Harrington", "given_name": "Leo A.", "clpid": "Harrington-L-A" }, { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We study in this paper characterizations of various interesting classes of relations arising in recursion theory. We first determine which Spector classes on the structure of arithmetic arise from recursion in normal type 2 objects, giving a partial answer to a problem raised by Moschovakis [8], where the notion of Spector class was first\nessentially introduced. Our result here was independently discovered by S. G. Simpson (see [3]). We conclude our study of Spector classes by examining two simple relations between them and a natural hierarchy to which they give rise.", "issn": "0022-4812", "publisher": "Association for Symbolic Logic", "publication": "Journal of Symbolic Logic", "publication_date": "1975-03", "series_number": "1", "volume": "40", "issue": "1", "pages": "19-24" }, { "id": "authors:qjq3h-yjz89", "collection": "authors", "collection_id": "qjq3h-yjz89", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130522-132343731", "type": "article", "title": "The Theory of Countable Analytical Sets", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "The purpose of this paper is the study of the structure of countable sets in the various levels of the analytical hierarchy of sets of reals. It is first shown that, assuming projective determinacy, there is for each odd n a largest countable \u220f_n^1 set of reals, C_n (this is also true for n even, replacing \u220f_n^1 by \u03a3_n^1 and has been established earlier by Solovay for n = 2 and by Moschovakis and the author for all even n > 2). The internal structure of the sets C_n is then investigated in detail, the point of departure being the fact that each C_n is a set of \u0394_n^1-degrees, wellordered under their usual partial ordering. Finally, a number of applications of the preceding theory is presented, covering a variety of topics such as specification of bases, \u03c9-models of analysis, higher-level analogs of the constructible universe, inductive definability, etc.", "issn": "0002-9947", "publisher": "American Mathematical Society", "publication": "Transactions of the American Mathematical Society", "publication_date": "1975-02", "series_number": "2", "volume": "202", "issue": "2", "pages": "259-297" }, { "id": "authors:hyf93-2sr27", "collection": "authors", "collection_id": "hyf93-2sr27", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:KECpjm75", "type": "article", "title": "Countable ordinals and the analytical hierarchy, I", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "The following results are proved, using the axiom of Projective Determinacy: (i) For n \u2265 1, every II(1/2n+1) set of countable ordinals contains a \u0394(1/2n+1) ordinal, (ii) For n \u2265 1, the set of reals \u0394(1/2n) in an ordinal is equal to the largest countable \u03a3(1/2n) set and (iii) Every real is \u0394(1/n) inside some transitive model of set theory if and only if n \u2265 4.", "issn": "0030-8730", "publisher": "Pacific Journal of Mathematics", "publication": "Pacific Journal of Mathematics", "publication_date": "1975", "series_number": "1", "volume": "60", "issue": "1", "pages": "223-227" }, { "id": "authors:8q2ev-n7z83", "collection": "authors", "collection_id": "8q2ev-n7z83", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130612-115543154", "type": "article", "title": "Countable ordinals and the analytical hierarchy. I.", "author": [ { "family_name": "Kechris", "given_name": "A. S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "The following results are proved, using the axiom of Projective Determinacy: (i) For n \u2ab4 1, every \u220f^1_(2n+1) set of countable ordinals contains a \u0394^1_(2n+1) ordinal, (ii) For n \u2ab4 1, the set of reals \u0394^1_(2n) in an ordinal is equal to the largest countable \u03a3^1_(2n) set and (iii) Every real is \u0394^1_n inside some transitive model of set theory if and only if n \u2ab4 4.", "issn": "0030-8730", "publisher": "Pacific Journal of Mathematics", "publication": "Pacific Journal of Mathematics", "publication_date": "1975", "series_number": "1", "volume": "60", "issue": "1", "pages": "223-227" }, { "id": "authors:k61kb-3by18", "collection": "authors", "collection_id": "k61kb-3by18", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130529-103426984", "type": "article", "title": "On Projective Ordinals", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We study in this paper the projective ordinals \u03b4^1_n, where \u03b4^1_n = sup{\u03be: \u03be is the length of \u0251 \u0394^1_n prewellordering of the continuum}. These ordinals were introduced by Moschovakis in [8] to serve as a measure of the \"definable length\" of the continuum. We prove first in \u00a72 that projective determinacy implies \u03b4^1_n < \u03b4^1_n for all even n > 0 (the same result for odd n is due to Moschovakis). Next, in the context of full determinacy, we partly generalize (in \u00a73) the classical fact that \u03b4^1_1 \u2135_l and the result of Martin that \u03b4^1_3 = \u2135_(\u03c9 + 1) by proving that \u03b4^1_(n2+1) = \u03bb^+_(2n+1), where \u03bb_(2n+1) is a cardinal of cofinality \u03c9. Finally we discuss in \u00a74 the connection between the projective ordinals and Solovay's uniform indiscernibles. We prove among other things that \u2200\u03b1 (\u03b1^# exists) implies that every \u03b4^1_n with n \u2265 3 is a fixed point of the increasing enumeration of the uniform indiscernibles.", "issn": "0022-4812", "publisher": "Association for Symbolic Logic", "publication": "Journal of Symbolic Logic", "publication_date": "1974-06", "series_number": "2", "volume": "39", "issue": "2", "pages": "269-282" }, { "id": "authors:ymvrw-ces91", "collection": "authors", "collection_id": "ymvrw-ces91", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130528-122128969", "type": "article", "title": "Measure and category in effective descriptive set theory", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" } ], "abstract": "We are concerned in this paper with some definability aspects of the theory of measure and category on the continuum. Our objects of study are the projective subsets of the reals and their structure from a measure theoretic and topological point of view.", "doi": "10.1016/0003-4843(73)90012-0", "issn": "0003-4843", "publisher": "Elsevier", "publication": "Annals of Mathematical Logic", "publication_date": "1973-07", "series_number": "4", "volume": "5", "issue": "4", "pages": "337-384" }, { "id": "authors:ekx2r-k0d03", "collection": "authors", "collection_id": "ekx2r-k0d03", "cite_using_url": "https://resolver.caltech.edu/CaltechAUTHORS:20130529-155104187", "type": "article", "title": "Two theorems about projective sets", "author": [ { "family_name": "Kechris", "given_name": "Alexander S.", "orcid": "0000-0002-2226-0423", "clpid": "Kechris-A-S" }, { "family_name": "Moschovakis", "given_name": "Yiannis N.", "clpid": "Moschovakis-Y-N" } ], "abstract": "In this paper we prove two (rather unrelated) theorems about projective sets. The first one asserts that subsets of \u2135_1 which are \u2211^1_2 in the codes are constructible; thus it extends the familiar theorem of Shoenfield that \u2211^1_2 subsets of \u03c9 are constructible. The second is concerned with largest countable \u2211^1_(2n) sets and establishes their existence under the hypothesis of Projective Determinacy and the assumption that there exist only countably many ordinal definable reals.", "doi": "10.1007/BF02764630", "issn": "0021-2172", "publisher": "Springer", "publication": "Israel Journal of Mathematics", "publication_date": "1972-12", "series_number": "4", "volume": "12", "issue": "4", "pages": "391-399" } ]