Journal of Symbolic Logic

Intrinsic bounds on complexity and definability at limit levels

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.

Article information

Source
J. Symbolic Logic Volume 74, Issue 3 (2009), 1047-1060.

Dates
First available in Project Euclid: 16 June 2009

https://projecteuclid.org/euclid.jsl/1245158098

Digital Object Identifier
doi:10.2178/jsl/1245158098

Mathematical Reviews number (MathSciNet)
MR2548479

Zentralblatt MATH identifier
1201.03019

Citation

Chisholm, John; Fokina, Ekaterina B.; Goncharov, Sergey S.; Harizanov, Valentina S.; Knight, Julia F.; Quinn, Sara. Intrinsic bounds on complexity and definability at limit levels. J. Symbolic Logic 74 (2009), no. 3, 1047--1060. doi:10.2178/jsl/1245158098. https://projecteuclid.org/euclid.jsl/1245158098