A question of Woodin asks if κ is strongly compact and GCH holds below κ, then must GCH hold everywhere? One variant of this question asks if κ is strongly compact and GCH fails at every regular cardinal δ < κ, then must GCH fail at some regular cardinal δ ≥ κ? Another variant asks if it is possible for GCH to fail at every limit cardinal less than or equal to a strongly compact cardinal κ. We ...

We show that a strong form of the so called Lindstrr om's Theorem 4] fails to generalize to extensions of L ! and L : For weakly compact there is no strongest extension of L ! with the (;)-compactness property and the LL owenheim-Skolem theorem down to. With an additional set-theoretic assumption, there is no strongest extension of L with the (;)-compactness property and the LL owenheim-Skolem ...

For α, β, n ordinals α → (β, n) means that any graph on the ordinal α either contains an independent set of order-type β or a complete subgraph of order-type n. We only consider cases where n is a natural number, hence the naming. The partition relation α → (β, n) for α, β finite multiples of κ where κ = ω or κ is weakly compact is characterized by a finitary problem as is the same for α, β fin...

We construct a model containing a proper class of strongly compact cardinals in which no strongly compact cardinal κ is κ supercompact and in which every strongly compact cardinal has its strong compactness resurrectible.

We examine what happens if we replace ZFC with a localistic/relativistic system, LZFC, whose central new axiom, denoted by Loc(ZFC), says that every set belongs to a transitive model of ZFC. LZFC consists of Loc(ZFC) plus some elementary axioms forming Basic Set Theory (BST). Some theoretical reasons for this shift of view are given. All Π2 consequences of ZFC are provable in LZFC. LZFC strongl...

Let  $left{a_alpharight}_{alphain I}$ be a bounded net in a Banach algebra $A$ and $varphi$ a nonzero multiplicative linear functional on $A$. In this paper, we deal with the problem of when $|aa_alpha-varphi(a)a_alpha|to0$ uniformly for all $a$ in weakly compact subsets of $A$. We show that Banach algebras associated to locally compact groups such  as Segal algebras and $L^1$-algebras are resp...

Let λ be an infinite cardinal number and let `∞(λ) denote the subspace of `∞(λ) consisting of all functions which assume at most countably many non zero values. We classify all infinite dimensional complemented subspaces of `∞(λ), proving that they are isomorphic to `∞(κ) for some cardinal number κ. Then we show that the Banach algebra of all bounded linear operators on `∞(λ) or `∞(λ) has the u...

Article history: Received 5 December 2006 Revised 21 April 2008 Available online 11 April 2009 A new axiomatic system OST of operational set theory is introduced in which the usual language of set theory is expanded to allow us to talk about (possibly partial) operations applicable both to sets and to operations. OST is equivalent in strength to admissible set theory, and a natural extension of...