TY - JOUR

T1 - Selection in the monadic theory of a countable ordinal

AU - Rabinovich, Alexander

AU - Shomrat, Amit

PY - 2008/9

Y1 - 2008/9

N2 - A monadic formula ψ(Y) is a selector for a formula φ(Y) in a structure M if 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 ≤ ω1we 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.

AB - A monadic formula ψ(Y) is a selector for a formula φ(Y) in a structure M if 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 ≤ ω1we 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.

UR - http://www.scopus.com/inward/record.url?scp=52049104970&partnerID=8YFLogxK

U2 - 10.2178/jsl/1230396747

DO - 10.2178/jsl/1230396747

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AN - SCOPUS:52049104970

SN - 0022-4812

VL - 73

SP - 783

EP - 816

JO - Journal of Symbolic Logic

JF - Journal of Symbolic Logic

IS - 3

ER -