Abstract not found

The results in this paper involve two different topics in the descriptive theory of Polish group actions. The book Becker-Kechris [6] is an introduction to that theory. Our two topics—and two collections of theorems—are rather unrelated, but the proofs for both topics are essentially the same.

iii In memory of my father. iv

1. Background on infinitary logic 2 1.1. Expressive power of Lω1ω 2 1.2. The back-and-forth construction 3

where ωM 1 = ωCK 1 and where the LωCK 1,ω-theory of M is not ω-categorical. The Scott rank of a model is the least ordinal β where the model is prime in its Lωβ,ω-theory. Most well-known models with unbounded-type; such a There exists a countable structure M of Scott rank ω CK 1 atoms below ω CK

Dedicated to the memory of S. Banach. Abstract. It is shown that any Banach space X of density dens(X) ≥ גω1 contains an (infinite) unconditional sequence. Here for any ordinal α the cardinal גα is given by induction: ג0 = ℵ0; גα+1 = exp(גα); גα = sup{גβ: β < α} if α is a limit ordinal, and ω1 is the first uncountable ordinal. Every Banach space of density dens(X) = τ ≥ ω1 contains an unconditional sequence of cardinality τ if and only if τ is a weakly compact cardinal. Other results that are concerned large cardinals and unconditional sequences are also presented. 1. Introduction. The known J. Lindenstrauss ’ problem [1] was: Whether every infinite dimensional Banach space contains an (infinite) unconditional sequence? It was successfully solved (in negative) by W.T. Gowers and B. Maurey [2].