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MUTUAL INTERPRETABILITY OF WEAK ESSENTIALLY UNDECIDABLE THEORIES

Part of: Mathematical Logic and Foundations Proof theory and constructive mathematics

Published online by Cambridge University Press:  18 February 2022

ZLATAN DAMNJANOVIC
ZLATAN DAMNJANOVIC*
Affiliation:
SCHOOL OF PHILOSOPHY UNIVERSITY OF SOUTHERN CALIFORNIA LOS ANGELES, CA90089, USA
*
E-mail:[email protected]
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Abstract

Kristiansen and Murwanashyaka recently proved that Robinson arithmetic, Q, is interpretable in an elementary theory of full binary trees, T. We prove that, conversely, T is interpretable in Q by producing a formal interpretation of T in an elementary concatenation theory QT+, thereby also establishing mutual interpretability of T with several well-known weak essentially undecidable theories of numbers, strings, and sets. We also introduce a “hybrid” elementary theory of strings and trees, WQT*, and establish its mutual interpretability with Robinson’s weak arithmetic R, the weak theory of trees WT of Kristiansen and Murwanashyaka, and the weak concatenation theory WTCε of Higuchi and Horihata.

Keywords

interpretabilityfull binary treesRobinson arithmeticconcatenation theorystringsessential undecidability

MSC classification

Primary: 03-02: Research exposition (monographs, survey articles)
Secondary: 03F25: Relative consistency and interpretations 03F30: First-order arithmetic and fragments
Type
Article
Information
The Journal of Symbolic Logic , Volume 87 , Issue 4 , December 2022 , pp. 1374 - 1395
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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