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ILLUSORY MODELS OF PEANO ARITHMETIC

Published online by Cambridge University Press:  13 July 2016

MAKOTO KIKUCHI  and
TAISHI KURAHASHI
MAKOTO KIKUCHI
Affiliation:
GRADUATE SCHOOL OF SYSTEM INFORMATICS KOBE UNIVERSITY 1-1 ROKKODAI, NADA KOBE 657-8501, JAPANE-mail: mkikuchi@kobe-u.ac.jp
TAISHI KURAHASHI
Affiliation:
DEPARTMENT OF NATURAL SCIENCES NATIONAL INSTITUTE OF TECHNOLOGY KISARAZU COLLEGE 2-11-1 KIYOMIDAI-HIGASHI KISARAZU CHIBA 292-0041, JAPANE-mail: kurahashi@nebula.n.kisarazu.ac.jp
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Abstract

By using a provability predicate of PA, we define ThmPA(M) as the set of theorems of PA in a model M of PA. We say a model M of PA is (1) illusory if ThmPA(M) ⊈ ThmPA(ℕ), (2) heterodox if ThmPA(M) ⊈ TA, (3) sane if M ⊨ ConPA, and insane if it is not sane, (4) maximally sane if it is sane and ThmPA(M) ⊆ ThmPA(N) implies ThmPA(M) = ThmPA(N) for every sane model N of PA. We firstly show that M is heterodox if and only if it is illusory, and that ThmPA(M) ∩ TA ≠ ThmPA(ℕ) for any illusory model M. Then we show that there exists a maximally sane model, every maximally sane model satisfies ¬ConPA+ConPA, and there exists a sane model of ¬ConPA+ConPA which is not maximally sane. We define that an insane model is (5) illusory by nature if its every initial segment being a nonstandard model of PA is illusory, and (6) going insane suddenly if its every initial segment being a sane model of PA is not illusory. We show that there exists a model of PA which is illusory by nature, and we prove the existence of a model of PA which is going insane suddenly.

Keywords

nonstandard models of arithmeticprovability predicatesGödel’s incompleteness theorems
Type
Articles
Information
The Journal of Symbolic Logic , Volume 81 , Issue 3 , September 2016 , pp. 1163 - 1175
Copyright
Copyright © The Association for Symbolic Logic 2016 

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