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ON SHAVRUKOV’S NON-ISOMORPHISM THEOREM FOR DIAGONALIZABLE ALGEBRAS

Part of: Proof theory and constructive mathematics

Published online by Cambridge University Press:  12 September 2022

EVGENY A. KOLMAKOV Open the ORCID record for EVGENY A. KOLMAKOV [Opens in a new window]
EVGENY A. KOLMAKOV*
Affiliation:
DEPARTMENT OF MATHEMATICAL LOGIC STEKLOV MATHEMATICAL INSTITUTE OF RUSSIAN ACADEMY OF SCIENCES 8 GUBKINA STREET, MOSCOW 119991, RUSSIA E-mail: kolmakov.evgn@gmail.com
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Abstract

We prove a strengthened version of Shavrukov’s result on the non-isomorphism of diagonalizable algebras of two $\Sigma _1$-sound theories, based on the improvements previously found by Adamsson. We then obtain several corollaries to the strengthened result by applying it to various pairs of theories and obtain new non-isomorphism examples. In particular, we show that there are no surjective homomorphisms from the algebra $(\mathfrak {L}_T, \Box _T\Box _T)$ onto the algebra $(\mathfrak {L}_T, \Box _T)$. The case of bimodal diagonalizable algebras is also considered. We give several examples of pairs of theories with isomorphic diagonalizable algebras but non-isomorphic bimodal diagonalizable algebras.

Keywords

provability predicatediagonalizable algebraprovability algebraShavrukov’s theoremisomorphismreflection principles

MSC classification

Primary: 03F30: First-order arithmetic and fragments
Secondary: 03F45: Provability logics and related algebras (e.g., diagonalizable algebras)

Information

Type
Research Article
Information
The Review of Symbolic Logic , Volume 17 , Issue 1 , March 2024 , pp. 206 - 243
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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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References

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