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Selection in the monadic theory of a countable ordinal

  • Alexander Rabinovich (a1) and Amit Shomrat (a2)


A monadic formula ψ(Y) is a selector for a formula φ(Y) in a structure if there exists a unique subset P of which satisfies ψ and this P also satisfies φ. We show that for every ordinal αωω there are formulas having no selector in the structure (α, <). For αω1, 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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Selection in the monadic theory of a countable ordinal

  • Alexander Rabinovich (a1) and Amit Shomrat (a2)


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