CaltechTHESIS advisor: Book Chapter
https://feeds.library.caltech.edu/people/Woodin-W-H/combined_advisor.rss
A Caltech Library Repository Feedhttp://www.rssboard.org/rss-specificationpython-feedgenenMon, 14 Oct 2024 13:44:31 -0700Independence Results for Indescribable Cardinals
https://resolver.caltech.edu/CaltechETD:etd-05222007-093217
Year: 1989
DOI: 10.7907/40tv-qe43
No abstract submitted.https://resolver.caltech.edu/CaltechETD:etd-05222007-093217Homogeneous sequences of cardinals for ordinal definable partition relations
https://resolver.caltech.edu/CaltechETD:etd-06132007-073731
Year: 1990
DOI: 10.7907/hchw-ca34
NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.
In 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.
Theorem 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. [...]
If 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 (*).
Moreover, 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 < [...].
Theorem 2. If [...] is [...] -generic over V(H) and V[...], then [...] + [...] strong limit + [...]. Therefore Con(ZFC + ([...]) [...] supercompact & [...] inaccessible & is [...]) [...] Con(ZFC + ([...] strong limit)[...]. [...]
https://resolver.caltech.edu/CaltechETD:etd-06132007-073731An Abstract Condensation Property
https://resolver.caltech.edu/CaltechTHESIS:11212011-135758866
Year: 1994
DOI: 10.7907/khcx-zk98
Let A = (A, ... ) be a relational structure. Say that A has condensation if there is an F : A^(ω) → 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
of 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
consequences. In particular, we show that if condensation holds in M, then M ╞ 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
that condensation implies ◊_κ(E) for κ regular and E κ 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
relationship 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
which hold in L, and an argument that Cond(G) fails in V[G], where G is generic for the partial order adding ω_2 cohen subsets of ω_1.https://resolver.caltech.edu/CaltechTHESIS:11212011-135758866