Journal of Symbolic Logic

Selection in the monadic theory of a countable ordinal

Alexander Rabinovich and Amit Shomrat

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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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J. Symbolic Logic, Volume 73, Issue 3 (2008), 783-816.

First available in Project Euclid: 27 December 2008

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Rabinovich, Alexander; Shomrat, Amit. Selection in the monadic theory of a countable ordinal. J. Symbolic Logic 73 (2008), no. 3, 783--816. doi:10.2178/jsl/1230396747.

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