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June 2013 A fixed point for the jump operator on structures
Antonio Montalbán
J. Symbolic Logic 78(2): 425-438 (June 2013). DOI: 10.2178/jsl.7802050

Abstract

Assuming that $0^\#$ exists, we prove that there is a structure that can effectively interpret its own jump. In particular, we get a structure $\mathcal A$ such that \[ \textit{Sp}({\mathcal A}) = \{{\bf x}'\colon {\bf x}\in \textit{Sp}({\mathcal A})\}, \] where $\textit{Sp}({\mathcal A})$ is the set of Turing degrees which compute a copy of $\mathcal A$. More interesting than the result itself is its unexpected complexity. We prove that higher-order arithmetic, which is the union of full $n$th-order arithmetic for all $n$, cannot prove the existence of such a structure.

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Antonio Montalbán. "A fixed point for the jump operator on structures." J. Symbolic Logic 78 (2) 425 - 438, June 2013. https://doi.org/10.2178/jsl.7802050

Information

Published: June 2013
First available in Project Euclid: 15 May 2013

zbMATH: 1305.03036
MathSciNet: MR3145189
Digital Object Identifier: 10.2178/jsl.7802050

Rights: Copyright © 2013 Association for Symbolic Logic

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Vol.78 • No. 2 • June 2013
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