Intrinsic bounds on complexity and definability at limit levels
John Chisholm, Ekaterina B. Fokina, Sergey S. Goncharov, Valentina S. Harizanov, Julia F. Knight, and Sara Quinn
Source: J. Symbolic Logic Volume 74, Issue 3
(2009), 1047-1060.
Abstract
We show that for every computable limit ordinal α, there is a computable structure 𝒜 that is Δα⁰ categorical, but not relatively Δα⁰ categorical (equivalently, it does not have a formally Σα⁰ Scott family). We also show that for every computable limit ordinal α, there is a computable structure 𝒜 with an additional relation R that is intrinsically Σα⁰ on 𝒜, but not relatively intrinsically Σα⁰ on 𝒜 (equivalently, it is not definable by a computable Σα formula with finitely many parameters). Earlier results in [7], [10], and [8] establish the same facts for computable successor ordinals α.
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Permanent link to this document: http://projecteuclid.org/euclid.jsl/1245158098
Digital Object Identifier: doi:10.2178/jsl/1245158098
Zentralblatt MATH identifier: 05609403
Mathematical Reviews number (MathSciNet): MR2548479
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