The motivation for this paper is the following: In [Fuc08] I showed that it is inconsistent with ZFC that the maximality principle for closed forcings holds at unboundedly many regular cardinals κ (even only allowing κ itself as a parameter in the maximality principle for <κ-closed forcings each time). So the question is whether it is consistent to have this principle at unboundedly many regula...

We prove that the following two statements are equiconsistent: there exists a greatly Mahlo cardinal; there exists a regular uncountable cardinal κ such that no stationary subset of κ+ ∩ cof(κ) carries a partial square. A famous theorem in set theory is the result that the failure of the square principle κ, for a regular uncountable cardinal κ, is equiconsistent with a Mahlo cardinal. Solovay p...

We prove the following: Theorem A. If D is a (λ+, κ)-regular ultrafilter, then either (a) D is (λ, κ)-regular, or (b) the cofinality of the linear order ∏ D〈λ, <〉 is cf κ, and D is (λ, κ′)-regular for all κ′ < κ. Corollary B. Suppose that κ is singular, κ > λ and either λ is regular, or cf κ < cf λ. Then every (λ+n, κ)-regular ultrafilter is (λ, κ)-regular. We also discuss some consequences and...

We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if V |= ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension V [G] |= ZFC + GCH in which, (a) (preservation) for κ ≤ λ regular, if V |...

We show that supercompactness and strong compactness can be equivalent even as properties of pairs of regular cardinals. Specifically, we show that if V |= ZFC + GCH is a given model (which in interesting cases contains instances of supercompactness), then there is some cardinal and cofinality preserving generic extension V [G] |= ZFC + GCH in which, (a) (preservation) for κ ≤ λ regular, if V |...

The following pcf results are proved: 1. Assume that κ > א0 is a weakly compact cardinal. Let μ > 2κ be a singular cardinal of cofinality κ. Then for every regular λ < pp+Γ(κ)(μ) there is an increasing sequence ⟨λi | i < κ⟩ of regular cardinals converging to μ such that λ = tcf( ∏ i<κ λi, <Jbd κ ). 2. Let μ be a strong limit cardinal and θ a cardinal above μ. Suppose that at least one of them h...

Via two short proofs and three constructions, we show how to increase the model-theoretic precision of a widely used method for building ultrafilters. We begin by showing that any flexible regular ultrafilter makes the product of an unbounded sequence of finite cardinals large, thus saturating any stable theory. We then prove directly that a “bottleneck” in the inductive construction of a regul...

We call a topological space κ-compact if every subset of size κ has a complete accumulation point in it. Let Φ(μ, κ, λ) denote the following statement: μ < κ < λ = cf(λ) and there is {Sξ : ξ < λ} ⊂ [κ] μ such that |{ξ : |Sξ ∩ A| = μ}| < λ whenever A ∈ [κ]. We show that if Φ(μ, κ, λ) holds and the space X is both μ-compact and λ-compact then X is κ-compact as well. Moreover, from PCF theory we d...

A Hausdorff topological group G is minimal if every continuous isomorphism f : G → H between G and a Hausdorff topological group H is open. Clearly, every compact Hausdorff group is minimal. It is well known that every infinite compact Hausdorff group contains a non-trivial convergent sequence. We extend this result to minimal abelian groups by proving that every infinite minimal abelian group ...

We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with certain additional “inner model like” properties. In particular, in this model, the class of Mahlo cardinals reflecting stationary sets is the same as the class of weakly compact cardinals, and every regular Jonsson cardinal is weakly compact. On the other hand, w...