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GENERALIZATIONS OF GÖDEL’S INCOMPLETENESS THEOREMS FOR ∑n-DEFINABLE THEORIES OF ARITHMETIC

Published online by Cambridge University Press:  07 November 2017

MAKOTO KIKUCHI  and
TAISHI KURAHASHI
MAKOTO KIKUCHI*
Affiliation:
Graduate School of System Informatics, Kobe University
TAISHI KURAHASHI*
Affiliation:
Department of Natural Science, National Institute of Technology Kisarazu College
*
*GRADUATE SCHOOL OF SYSTEM INFORMATICS KOBE UNIVERSITY 1-1 ROKKODAI, NADA, KOBE 657-8501, JAPAN E-mail: mkikuchi@kobe-u.ac.jp
†DEPARTMENT OF NATURAL SCIENCE NATIONAL INSTITUTE OF TECHNOLOGY, KISARAZU COLLEGE 2-11-1 KIYOMIDAI-HIGASHI, KISARAZU, CHIBA 292-0041, JAPAN E-mail: kurahashi@n.kisarazu.ac.jp
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Abstract

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It is well known that Gödel’s incompleteness theorems hold for ∑1-definable theories containing Peano arithmetic. We generalize Gödel’s incompleteness theorems for arithmetically definable theories. First, we prove that every ∑n+1-definable ∑n-sound theory is incomplete. Secondly, we generalize and improve Jeroslow and Hájek’s results. That is, we prove that every consistent theory having ∏n+1 set of theorems has a true but unprovable ∏n sentence. Lastly, we prove that no ∑n+1-definable ∑n -sound theory can prove its own ∑n-soundness. These three results are generalizations of Rosser’s improvement of the first incompleteness theorem, Gödel’s first incompleteness theorem, and the second incompleteness theorem, respectively.

Keywords

03F3003F40Gödel’s incompleteness theorems∑n-definable theories
Type
Research Article
Information
The Review of Symbolic Logic , Volume 10 , Issue 4 , December 2017 , pp. 603 - 616
Copyright
Copyright © Association for Symbolic Logic 2017 

References

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