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Polish group actions: dichotomies and generalized elementary embeddings
 J. Amer. Math. Soc
, 1997
"... The results in this paper involve two different topics in the descriptive theory of Polish group actions. The book BeckerKechris [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. ..."
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The results in this paper involve two different topics in the descriptive theory of Polish group actions. The book BeckerKechris [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.
BARWISE: INFINITARY LOGIC AND ADMISSIBLE SETS
"... 1. Background on infinitary logic 2 1.1. Expressive power of Lω1ω 2 1.2. The backandforth construction 3 ..."
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1. Background on infinitary logic 2 1.1. Expressive power of Lω1ω 2 1.2. The backandforth construction 3
Atomic models higher up
, 2008
"... There exists a countable structure M of Scott rank ωCK 1 where ωM 1 = ωCK 1 and where the LωCK 1,ωtheory of M is not ωcategorical. The Scott rank of a model is the least ordinal β where the model is prime in its Lωβ,ωtheory. Most wellknown models with unboundedtype; such a atoms below ω CK 1 al ..."
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There exists a countable structure M of Scott rank ωCK 1 where ωM 1 = ωCK 1 and where the LωCK 1,ωtheory of M is not ωcategorical. The Scott rank of a model is the least ordinal β where the model is prime in its Lωβ,ωtheory. Most wellknown models with unboundedtype; such a atoms below ω CK 1 also realize a nonprincipal L ω CK 1,ω will have Scott rank ωCK 1 + 1. Makkai ([4]) produces a hyperarithmetical model of Scott rank ωCK 1 whose LωCK 1,ωtheory is ωcategorical. A computable variant of Makkai’s example is produced in [2]. model that preserves the Σ1admissibility of ω CK 1
Index sets for computable structures
 Algebra and Logic
"... The index set of a computable structure A is the set of indices for computable copies of A. We determine the complexity of the index sets of various mathematically interesting structures, including arbitrary finite structures, Qvector spaces, Archimedean real closed ordered fields, reduced Abelian ..."
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The index set of a computable structure A is the set of indices for computable copies of A. We determine the complexity of the index sets of various mathematically interesting structures, including arbitrary finite structures, Qvector spaces, Archimedean real closed ordered fields, reduced Abelian pgroups of length less than ω 2, and models of the original Ehrenfeucht theory. The index sets for these structures all turn out to be mcomplete Π 0 n, dΣ 0 n,orΣ 0 n, for various n. In each case, the calculation involves finding an “optimal ” sentence (i.e., one of simplest form) that describes the structure. The form of the sentence (computable Πn, dΣn, or Σn) yields a bound on the complexity of the index set. When we show mcompleteness of the index set, we know that the sentence is optimal. For some structures, the first sentence that comes to mind is not optimal, and another sentence of simpler form is shown to serve the purpose. For some of the groups, this involves Ramsey theory.
On Banach Spaces of Large Density Character and Unconditional Sequences
, 2002
"... 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 t ..."
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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].