Starting from a supercompact cardinal κ, we force and construct a model in which κ is both the least strongly compact and least supercompact cardinal and κ’s strong compactness, but not its supercompactness, is indestructible under arbitrary κ-directed closed forcing.

By using combinatorial properties of stationary sets, we give a simple proof of some generalization of Silver’s Theorem i.e. if κ is an uncountable regular cardinal such that אκ is a singular strong limit cardinal, then the following hold. (1). If {α < κ : א<κ α ≤ אα·2} is stationary, then 2אκ ≤ אκ·2. (2). If {α < κ : א<κ α ≤ אα+γ}, where 0 < γ < κ, is stationary, then 2אκ ≤ אκ+γ.

A classical theorem of set theory is the equivalence of the weak square principle μ with the existence of a special Aronszajn tree on μ +. We introduce the notion of a weak square sequence on any regular uncountable cardinal, and prove that the equivalence between weak square sequences and special Aronszajn trees holds in general. Recall the weak square principle μ for an infinite cardinal μ, w...

In this paper, we show that for every locally compact abelian group G, the following statements are equivalent: (i) G contains no sequence {xn}n=0 such that {0} ∪ {±xn | n ∈ N} is infinite and quasiconvex in G, and xn −→ 0; (ii) one of the subgroups {g ∈ G | 2g = 0} and {g ∈ G | 3g = 0} is open in G; (iii) G contains an open compact subgroup of the form Z2 or Z κ 3 for some cardinal κ.

We answer some question of [Gi]. The upper bound of [Gi] on the strength of N S µ + precipitous for a regular µ is proved to be exact. It is shown that saturatedness of N S ℵ 0 κ over inaccessible κ requires at least o(κ) = κ ++. The upper bounds on the strength of N S κ precipitous for inaccessible κ are reduced quite close to the lower bounds.

Given a supercompact cardinal κ and a regular cardinal λ < κ, we describe a type of forcing such that in the generic extension the cofinality of κ is λ, there is a very good scale at κ, a bad scale at κ, and SCH at κ fails. When creating our model we have great freedom in assigning the value of 2, and so we can make SCH hold or fail arbitrarily

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.

A method of iteration of Prikry type forcing notions as well as a forcing for adding clubs is presented. It is applied to construct a model with a measurable cardinal containing a club of former regulars, starting with o(κ) = κ+ 1. On the other hand, it is shown that the strength of above is at least o(κ) = κ. Suppose that κ is an inaccessible cardinal. We wish to find a generic extension (usua...

Let κ < λ be regular uncountable cardinals. Using a finite support iteration of ccc posets we obtain the consistency of b = a = κ < s = λ. If μ is a measurable cardinal and μ < κ < λ, then using similar techniques we obtain the consistency of b = κ < a = s = λ.