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June 2007 Local definitions in degree structures: the Turing jump, hyperdegrees and beyond
Richard A. Shore
Bull. Symbolic Logic 13(2): 226-239 (June 2007). DOI: 10.2178/bsl/1185803806

Abstract

There are $\Pi_5$ormulas in the language of the Turing degrees, $\mathcal{D}$ with $\leq, \lor$ and $\land$ that define the relations $\mathbf{x}^{\prime\prime} \leq \mathbf{y}^{\prime\prime}, \mathbf{x}^{\prime\prime}=\mathbf{y}^{\prime\prime}$and so $\mathbf{x} \in \mathbf{L}_{2}(\rm{y}) = {\mathbf{x} \geq \mathbf{y}\, \vert \, \mathbf{x}^{\prime\prime}= \mathbf{y}^{\prime\prime}}$ in any jump ideal containing $\mathbf{0}^{\omega}$. There are also $\Sigma_6 $& $\Pi_6$ and $\Pi_8$ formulas that define the relations $\mathbb{w} = \mathbb{x}^{\prime\prime}$ and $\mathbb{w} = \mathbb{x}^{\prime}$, respectively, in any such ideal $\mathcal{I}$. In the language with just $\leq$ the quantifier complexity of each of these definitions increases by one. On the other hand, no $\Pi_2$ or $\Sigma_2$ formula in the language with just $\leq$ defines $\mathbf{L}_2$ or $\mathbf{x} \in \mathbf{L}_2(\mathbf{y})$. Our arguments and constructions are purely degree theoretic without any appeals to absoluteness considerations, set theoretic methods or coding of models of arithmetic. As a corollary, we see that every automorphism of $\mathcal{I}$ is fixed on every degree above $\mathbf{0}^{\prime\prime}$ and every relation on I that is invariant under double jump or joining with $\mathbf{0}^{\prime\prime}$ is definable over I if and only if it is definable in second order arithmetic with set quantification ranging over sets whose degrees are in $\mathcal{I}$ Similar direct coding arguments show that every hyperjump ideal I is rigid and biinterpretable with second order arithmetic with set quantification ranging over sets with hyperdegrees in $\mathcal{I}$. Analogous results hold for various coarser degree structures.

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Richard A. Shore. "Local definitions in degree structures: the Turing jump, hyperdegrees and beyond." Bull. Symbolic Logic 13 (2) 226 - 239, June 2007. https://doi.org/10.2178/bsl/1185803806

Information

Published: June 2007
First available in Project Euclid: 30 July 2007

zbMATH: 1131.03018
MathSciNet: MR2323843
Digital Object Identifier: 10.2178/bsl/1185803806

Rights: Copyright © 2007 Association for Symbolic Logic

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Vol.13 • No. 2 • June 2007
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