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PRENEX NORMAL FORM THEOREMS IN SEMI-CLASSICAL ARITHMETIC

Part of: General logic Proof theory and constructive mathematics

Published online by Cambridge University Press:  11 June 2021

MAKOTO FUJIWARA  and
TAISHI KURAHASHI
MAKOTO FUJIWARA
Affiliation:
SCHOOL OF SCIENCE AND TECHNOLOGY MEIJI UNIVERSITY, 1-1-1 HIGASHI-MITA, TAMA-KU, KAWASAKI-SHI214-8571KANAGAWA, JAPANE-mail:makotofujiwara@meiji.ac.jp
TAISHI KURAHASHI
Affiliation:
GRADUATE SCHOOL OF SYSTEM INFORMATICS KOBE UNIVERSITY, 1-1 ROKKODAI, NADA 657-8501 KOBE, JAPANE-mail:kurahashi@people.kobe-u.ac.jp
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Abstract

Akama et al. [1] systematically studied an arithmetical hierarchy of the law of excluded middle and related principles in the context of first-order arithmetic. In that paper, they first provide a prenex normal form theorem as a justification of their semi-classical principles restricted to prenex formulas. However, there are some errors in their proof. In this paper, we provide a simple counterexample of their prenex normal form theorem [1, Theorem 2.7], then modify it in an appropriate way which still serves to largely justify the arithmetical hierarchy. In addition, we characterize a variety of prenex normal form theorems by logical principles in the arithmetical hierarchy. The characterization results reveal that our prenex normal form theorems are optimal. For the characterization results, we establish a new conservation theorem on semi-classical arithmetic. The theorem generalizes a well-known fact that classical arithmetic is $\Pi _2$ -conservative over intuitionistic arithmetic.

Keywords

Prenex normal form theoremsemi-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 86 , Issue 3 , September 2021 , pp. 1124 - 1153
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Copyright
© Association for Symbolic Logic 2021

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References

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