Under the assumption that δ is a Woodin cardinal and GCH holds, I show that if F is any class function from the regular cardinals to the cardinals such that (1) κ < cf(F (κ)), (2) κ < λ implies F (κ) ≤ F (λ), and (3) δ is closed under F , then there is a cofinality-preserving forcing extension in which 2 = F (γ) for each regular cardinal γ < δ, and in which δ remains Woodin. Unlike the analogou...

We show that Shelah’s Eventual Categoricity Conjecture follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC with LS(K) below a strongly compact cardinal κ is < κ tame and applying the categoricity transfer of Grossberg and VanDieren [GV06a]. These techniques also apply t...

I investigate versions of the Maximality Principles for the classes of forcings which are <κ-closed, <κ-directed-closed, or of the form Col(κ,< λ). These principles come in many variants, depending on the parameters which are allowed. I shall write MPΓ(A) for the maximality principle for forcings in Γ, with parameters from A. The main results of this paper are: • The principles have many conseq...

We strengthen the revised GCH theorem by showing, e.g., that for λ = cf(λ) > iω, for all but finitely many regular κ < iω, it holds that “λ is accessible on cofinality κ” in some weak sense (see below). As a corollary, λ = 2 = μ > iω implies that the diamond holds on λ when restricted to cofinality κ for all but finitely many κ ∈ Reg∩iω. We strengthen previous results on the black box and the m...

Strongly Unfoldable Cardinals Made Indestructible by Thomas A. Johnstone Advisor: Joel David Hamkins I provide indestructibility results for weakly compact, indescribable and strongly unfoldable cardinals. In order to make these large cardinals indestructible, I assume the existence of a strongly unfoldable cardinal κ, which is a hypothesis consistent with V = L. The main result shows that any ...

We show that if cov(M) = κ, where κ is a regular cardinal such that ∀λ < κ(2 ≤ κ), then for every unbounded directed family H of size κ there is an ultrafilter UH such that the relativized Mathias forcing M(UH) preserves the unboundedness of H. This improves a result of M. Canjar (see [4, Theorem 10]). We discuss two instances of generic ultrafilters for which the relativized Mathias forcing pr...

We show that if cov(M) = κ, where κ is a regular cardinal such that ∀λ < κ(2 ≤ κ), then for every unbounded directed family H of size κ there is an ultrafilter UH such that the relativized Mathias forcing M(UH) preserves the unboundedness of H. This improves a result of M. Canjar (see [4, Theorem 10]). We discuss two instances of generic ultrafilters for which the relativized Mathias forcing pr...

A topological space X is C-κ-normal (C-mildly normal ) if there exist a κ-normal (mildly normal) Y and bijective function f : → such that the restriction f|A A→ f(A) homeomorphism for each compact subspace ⊆ X. We present new results about those two properties use discrete extension to solve open problems regarding C2-paracompactness α-normality

In this note we sketch the proofs of two results in combinatorial set theory. The common theme of the results is singular cardinal combinatorics: they involve the interaction between forcing, large cardinals, PCF theory and versions of Jensen’s weak square principle. 1. Changing cofinality Our first result involves an old question about changes of cofinality. Suppose that V,W are inner models o...

We prove a general theorem which includes most notions of “exact completion” as special cases. The theorem is that “κ-ary exact categories” are a reflective sub-2-category of “κ-ary sites”, for any regular cardinal κ. A κ-ary exact category is an exact category with disjoint and universal κ-small coproducts, and a κ-ary site is a site whose covering sieves are generated by κ-small families and ...