[
{
"id": "thesis:6736",
"collection": "thesis",
"collection_id": "6736",
"cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:11212011-135758866",
"type": "thesis",
"title": "An Abstract Condensation Property",
"author": [
{
"family_name": "Law",
"given_name": "David Richard",
"clpid": "Law-David Richard-Mathematics"
}
],
"thesis_advisor": [
{
"family_name": "Woodin",
"given_name": "W. Hugh",
"clpid": "Woodin-W-H"
}
],
"thesis_committee": [
{
"family_name": "Unknown",
"given_name": "Unknown"
}
],
"local_group": [
{
"literal": "div_pma"
}
],
"abstract": "Let A = (A, ... ) be a relational structure. Say that A has condensation if there is an F : A^(\u03c9) \u2192 A such that for every partial order P, it is forced by P that substructures of P which are closed under F are isomorphic to elements\r\nof the ground model. Condensation holds if every structure in V, the universe of sets, has condensation. This property, isolated by Woodin, captures part of the content of the condensation lemmas for L, K and other \"L-like\" models. We present a variety of results having to do with condensation in this abstract sense. Section 1 establishes the absoluteness of condensation and some of its\r\nconsequences. In particular, we show that if condensation holds in M, then M \u255e GCH and there are no measurable cardinals or precipitous ideals in M. The results of this section are due to Woodin. Section 2 contains a proof\r\nthat condensation implies \u25ca_\u03ba(E) for \u03ba regular and E \u03ba stationary. This is the main result of this thesis. The argument provides a new proof of the key lemma giving GCH. Section 2 also contains some information about the\r\nrelationship between condensation and strengthenings of diamond. Section 3 contains partial results having to do with forcing \"Cond(A)\", some further discussion of the relation between condensation and combinatorial principles\r\nwhich hold in L, and an argument that Cond(G) fails in V[G], where G is generic for the partial order adding \u03c9_2 cohen subsets of \u03c9_1.",
"doi": "10.7907/khcx-zk98",
"publication_date": "1994",
"thesis_type": "phd",
"thesis_year": "1994"
},
{
"id": "thesis:2572",
"collection": "thesis",
"collection_id": "2572",
"cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-06132007-073731",
"primary_object_url": {
"basename": "Kafkoulis_g_1990.pdf",
"content": "final",
"filesize": 3253446,
"license": "other",
"mime_type": "application/pdf",
"url": "/2572/1/Kafkoulis_g_1990.pdf",
"version": "v3.0.0"
},
"type": "thesis",
"title": "Homogeneous sequences of cardinals for ordinal definable partition relations",
"author": [
{
"family_name": "Kafkoulis",
"given_name": "George",
"clpid": "Kafkoulis-G"
}
],
"thesis_advisor": [
{
"family_name": "Kechris",
"given_name": "Alexander S.",
"clpid": "Kechris-A-S"
},
{
"family_name": "Woodin",
"given_name": "W. Hugh",
"clpid": "Woodin-W-H"
}
],
"thesis_committee": [
{
"family_name": "Unknown",
"given_name": "Unknown"
}
],
"local_group": [
{
"literal": "div_pma"
}
],
"abstract": "NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.\n\nIn this dissertation we study the consistency strength of the theory ZFC & ([...] strong limit)([...] < [...])([...]) (*), and we prove the consistency of this theory relative to the consistency of the existence of a supercompact cardinal and an inaccessible above it. If U is a normal measure on [...], then [...] denotes the Supercompact Prikry forcing induced by U. [...] is the partition relation [...] except that we consider only OD colorings of [...]. Theorems 1,2 are the main results of our thesis.\n\nTheorem 1. If there exists a model of ZFC in which [...] is a supercompact cardinal and [...] is an innaccessible above [...], then we can construct a model V of the same properties with the additional property that if U is a normal [...]-measure and G is [...] - generic over V, then V[G] does not satisfy the [...] partition property. [...]\n\nIf G is a [...]-generic over V filter, then we define H to be the set H: [...], and we consider the inner model V(H), which is the smallest inner model of ZF that contains H as an element. We prove that V(H) satisfies the above partition property (*).\n\nMoreover, V(H) satisfies < [...] - DC and using this fact we define a forcing [...], which is almost-homogeneous, < [...] - closed forcing that forces the AC over V(H) and does not add any new sets of rank < [...].\n\nTheorem 2. If [...] is [...] -generic over V(H) and V[...], then [...] + [...] strong limit + [...]. Therefore Con(ZFC + ([...]) [...] supercompact & [...] inaccessible & is [...]) [...] Con(ZFC + ([...] strong limit)[...]. [...]\n",
"doi": "10.7907/hchw-ca34",
"publication_date": "1990",
"thesis_type": "phd",
"thesis_year": "1990"
},
{
"id": "thesis:1949",
"collection": "thesis",
"collection_id": "1949",
"cite_using_url": "https://resolver.caltech.edu/CaltechETD:etd-05222007-093217",
"type": "thesis",
"title": "Independence Results for Indescribable Cardinals",
"author": [
{
"family_name": "Hauser",
"given_name": "Kai",
"clpid": "Hauser-Kai"
}
],
"thesis_advisor": [
{
"family_name": "Woodin",
"given_name": "W. Hugh",
"clpid": "Woodin-W-H"
}
],
"thesis_committee": [
{
"family_name": "Woodin",
"given_name": "W. Hugh",
"clpid": "Woodin-W-H"
},
{
"family_name": "Luxemburg",
"given_name": "W. A. J.",
"clpid": "Luxemburg-W-A-J"
},
{
"family_name": "Kechris",
"given_name": "Alexander S.",
"clpid": "Kechris-A-S"
},
{
"family_name": "Laver",
"given_name": "Richard",
"clpid": "Laver-Richard"
}
],
"local_group": [
{
"literal": "div_pma"
}
],
"abstract": "No abstract submitted.",
"doi": "10.7907/40tv-qe43",
"publication_date": "1989",
"thesis_type": "phd",
"thesis_year": "1989"
}
]