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Generalized reduction theorems for model-theoretic analogs of the class of coanalytic sets
Published online by Cambridge University Press: 12 March 2014
Abstract
Let 
be an admissible set. A sentence of the form 
is a 
sentence if φ ∈ (φ is ∨ Φ where Φ is an -r.e. set of sentences from 
). A sentence of the form 
is an 
, sentence if φ is a sentence. A class of structures is, for example, a ∀1 class if it is the class of models of a ∀1() sentence. Thus ∀1() is a class of classes of structures, and so forth.
Let 
i, be the structure 〈i, <〉, for i > 0. Let Γ be a class of classes of structures. We say that a sequence J1, …, Ji,…, i < ω, of classes of structures is a Γ sequence if Ji ∈ Γ, i < ω, and there is I ∈ Γ such that 
∈ Ji, if and only if [],i, where [,] is the disjoint sum. A class Γ of classes of structures has the easy uniformization property if for every Γ sequence J1,…, Ji,…, i < ω, there is a Γ sequence J′t, …, J′i, …, i < ω, such that J′i ⊆ Ji, i < ω, ⋃J′i = ⋃Ji, and the J′i are pairwise disjoint. The easy uniformization property is an effective version of Kuratowski's generalized reduction property that is closely related to Moschovakis's (topological) easy uniformization property.
We show over countable structures that ∀1() and ∃2() have the easy uniformization property if is a countable admissible set with an infinite member, that 
and 
have the easy uniformization property if α is countable, admissible, and not weakly stable, and that 
and 
have the easy uniformization properly. The results proved are more general. The result for answers a question of Vaught(1980).
- Type
- Research Article
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- Copyright © Association for Symbolic Logic 1993
References
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