### Selection in the monadic theory of a countable ordinal

Alexander Rabinovich and Amit Shomrat
Source: J. Symbolic Logic Volume 73, Issue 3 (2008), 783-816.

#### Abstract

A monadic formula ψ(Y) is a selector for a formula φ(Y) in a structure ℳ if there exists a unique subset P of ℳ which satisfies ψ and this P also satisfies φ. We show that for every ordinal α≥ ωω there are formulas having no selector in the structure (α, <). For α ≤ ω1, we decide which formulas have a selector in (α, <), and construct selectors for them. We deduce the impossibility of a full generalization of the Büchi-Landweber solvability theorem from (ω, <) to (ωω, <). We state a partial extension of that theorem to all countable ordinals. To each formula we assign a selection degree which measures “how difficult it is to select”. We show that in a countable ordinal all non-selectable formulas share the same degree.

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Permanent link to this document: http://projecteuclid.org/euclid.jsl/1230396747
Digital Object Identifier: doi:10.2178/jsl/1230396747
Mathematical Reviews number (MathSciNet): MR2444268
Zentralblatt MATH identifier: 1163.03013

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