In this paper we complete the characterization of Ext(G, Z) for any torsion-free abelian group G assuming Gödel’s axiom of constructibility plus there is no weakly compact cardinal. In particular, we prove in (V = L) that, for a singular cardinal ν of uncountable cofinality which is less than the first weakly compact cardinal and for every sequence of cardinals (νp : p ∈ Π) satisfying νp ≤ 2 , ...

Let B be a complete Boolean algebra. We show that if λ is an infinite cardinal and B is weakly (λ, ω)-distributive, then B is (λ, 2)-distributive. Using a similar argument, we show that if κ is a weakly compact cardinal such that B is weakly (2, κ)distributive and B is (α, 2)-distributive for each α < κ, then B is (κ, 2)-distributive.

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 ...

Given an uncountable regular cardinal κ, a partial order is κstationarily layered if the collection of regular suborders of P of cardinality less than κ is stationary in Pκ(P). We show that weak compactness can be characterized by this property of partial orders by proving that an uncountable regular cardinal κ is weakly compact if and only if every partial order satisfying the κ-chain conditio...

We extract some properties of Mahlo’s operation and show that some other very natural operations share these properties. The weakly compact sets form a similar hierarchy as the stationary sets. The height of this hierarchy is a large cardinal property connected to saturation properties of the weakly compact ideal. 1 2

We construct a model in which there are no @n-Aronszajn trees for any nite n 2, starting from a model with innnitely many supercompact cardinals. We also construct a model in which there is no ++-Aronszajn tree for a strong limit cardinal of coonality !, starting from a model with a supercompact cardinal and a weakly compact cardinal above it.

The Necessary Maximality Principle for c.c.c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal. The Necessary Maximality Principle for c.c.c. forcing, denoted 2mpccc(R), asserts that any statement about a real in a c.c.c. extension that could become true in a further c.c.c. extension and remain true in all subsequent c.c.c. extensions, is already tr...

The Necessary Maximality Principle for c.c.c. forcing with real parameters is equiconsistent with the existence of a weakly compact cardinal. The Necessary Maximality Principle for c.c.c. forcing, denoted 2mpccc(R), asserts that any statement about a real in a c.c.c. extension that could become true in a further c.c.c. extension and remain true in all subsequent c.c.c. extensions, is already tr...

We show that the consistency strength of the system NFUB, recently introduced by Randall Holmes, is precisely that of ZFC − + “There is a weakly compact cardinal”.