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A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenTue, 16 Apr 2024 13:26:38 +0000The countable admissible ordinal equivalence relation
https://resolver.caltech.edu/CaltechAUTHORS:20170515-084610472
Authors: {'items': [{'id': 'Chan-William', 'name': {'family': 'Chan', 'given': 'William'}}]}
Year: 2017
DOI: 10.1016/j.apal.2016.12.002
Let F_(ω1) be the countable admissible ordinal equivalence relation defined on ^ω2 by x F_(ω1) y if and only if ω_1^x=ω_1^y. Some invariant descriptive set theoretic properties of F_(ω1) will be explored using infinitary logic in countable admissible fragments as the main tool. Marker showed F_(ω1) is not the orbit equivalence relation of a continuous action of a Polish group on ^ω2. Becker stengthened this to show F_(ω1) is not even the orbit equivalence relation of a Δ_1^1 action of a Polish group. However, Montalbán has shown that F_(ω1) is Δ_1^1 reducible to an orbit equivalence relation of a Polish group action, in fact, F_(ω1) is classifiable by countable structures. It will be shown here that F_(ω1) must be classified by structures of high Scott rank. Let E_(ω1) denote the equivalence of order types of reals coding well-orderings. If E and F are two equivalence relations on Polish spaces X and Y, respectively, E ≤ aΔ_1^1 F denotes the existence of a Δ_1^1 function f:X→Y which is a reduction of E to F, except possibly on countably many classes of E. Using a result of Zapletal, the existence of a measurable cardinal implies E_(ω1) ≤ aΔ_1^1 F_(ω1). However, it will be shown that in Gödel's constructible universe L (and set generic extensions of L), E_(ω1) ≤ aΔ_1^1 F_(ω1) is false. Lastly, the techniques of the previous result will be used to show that in L (and set generic extensions of L), the isomorphism relation induced by a counterexample to Vaught's conjecture cannot be Δ_1^1 reducible to F_(ω1). This shows the consistency of a negative answer to a question of Sy-David Friedman.https://authors.library.caltech.edu/records/hw1tn-yqg76Equivalence Relations Which Are Borel Somewhere
https://resolver.caltech.edu/CaltechAUTHORS:20171002-095042985
Authors: {'items': [{'id': 'Chan-William', 'name': {'family': 'Chan', 'given': 'William'}}]}
Year: 2017
DOI: 10.1017/jsl.2017.22
The following will be shown: Let I be a σ-ideal on a Polish space X so that the associated forcing of I + Δ^1_1 sets ordered by ⊆ is a proper forcing. Let E be a Σ^1_1 or a ∏^1_1 equivalence relation on X with all equivalence classes Δ^1_1. If for all z ∈_H(2^N0)+, z ♯ exists, then there exists an I + Δ^1_1 set C ⊆ X such that E ↾ C is a Δ^1_1 equivalence relation.https://authors.library.caltech.edu/records/5nn8x-35a94