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Countable structures, Ehrenfeucht strategies, and Wadge reductions

Published online by Cambridge University Press:  12 March 2014

Tom Linton
Tom Linton*
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, Wisconsin 53706
*
Department of Mathematics, California Institute of Technology, Pasadena, California 91125.
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Abstract

For countable structures and , let abbreviate the statement that every sentence true in also holds in . One can define a back and forth game between the structures and that determines whether . We verify that if θ is an Lω,ω sentence that is not equivalent to any Lω,ω sentence, then there are countably infinite models and such that ⊨ θ, ⊨ ¬θ, and . For countable languages ℒ there is a natural way to view ℒ structúres with universe ω as a topological space, Xℒ. Let [] = { ∊ Xℒ∣ ≅ } denote the isomorphism class of . Let and be countably infinite nonisomorphic ℒ structures, and let C ⊆ ωω be any subset. Our main result states that if , then there is a continuous function f: ωω → Xℒ with the property that x ∊ C ⇒ f(x) ∊ [] and x ∉ C ⇒ f(x) ∊ f(x) ∈ []. In fact, for α ≤ 3, the continuous function f can be defined from the relation.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 56 , Issue 4 , December 1991 , pp. 1325 - 1348
Copyright
Copyright © Association for Symbolic Logic 1991

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