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

  • MAKOTO KIKUCHI (a1) and TAISHI KURAHASHI (a2)
Abstract
Abstract

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.

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*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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