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.

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...