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CONSERVATION THEOREMS ON SEMI-CLASSICAL ARITHMETIC

Part of: General logic Proof theory and constructive mathematics

Published online by Cambridge University Press:  17 March 2022

MAKOTO FUJIWARA Open the ORCID record for MAKOTO FUJIWARA [Opens in a new window]  and
TAISHI KURAHASHI Open the ORCID record for TAISHI KURAHASHI [Opens in a new window]
MAKOTO FUJIWARA*
Affiliation:
SCHOOL OF SCIENCE AND TECHNOLOGY MEIJI UNIVERSITY 1-1-1 HIGASHI-MITA, TAMA-KU, KAWASAKI-SHI KANAGAWA 214-8571, JAPAN
TAISHI KURAHASHI
Affiliation:
GRADUATE SCHOOL OF SYSTEM INFORMATICS KOBE UNIVERSITY 1-1 ROKKODAI, NADA, KOBE 657-8501, JAPAN E-mail: [email protected]
*
E-mail: [email protected]
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Abstract

We systematically study conservation theorems on theories of semi-classical arithmetic, which lie in-between classical arithmetic $\mathsf {PA}$ and intuitionistic arithmetic $\mathsf {HA}$. Using a generalized negative translation, we first provide a structured proof of the fact that $\mathsf {PA}$ is $\Pi _{k+2}$-conservative over $\mathsf {HA} + {\Sigma _k}\text {-}\mathrm {LEM}$ where ${\Sigma _k}\text {-}\mathrm {LEM}$ is the axiom scheme of the law-of-excluded-middle restricted to formulas in $\Sigma _k$. In addition, we show that this conservation theorem is optimal in the sense that for any semi-classical arithmetic T, if $\mathsf {PA}$ is $\Pi _{k+2}$-conservative over T, then ${T}$ proves ${\Sigma _k}\text {-}\mathrm {LEM}$. In the same manner, we also characterize conservation theorems for other well-studied classes of formulas by fragments of classical axioms or rules. This reveals the entire structure of conservation theorems with respect to the arithmetical hierarchy of classical principles.

Keywords

conservation theoremssemi-classical arithmeticintuitionistic arithmetic

MSC classification

Primary: 03B20: Subsystems of classical logic (including intuitionistic logic)
Secondary: 03F03: Proof theory, general 03F30: First-order arithmetic and fragments 03F50: Metamathematics of constructive systems
Type
Article
Information
The Journal of Symbolic Logic , Volume 88 , Issue 4 , December 2023 , pp. 1469 - 1496
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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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