Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-27T04:11:59.582Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Craig, W., On axiomatizability within a system, this Journal, vol. 18 (1953), pp. 30–32.Google Scholar
Guaspari, D. and Solovay, R. M., Rosser sentences . Annals of Mathematical Logic, vol. 16 (1979), no. 1, pp. 81–99.CrossRefGoogle Scholar
Hájek, P., On a new notion of partial conservativity , Computation and Proof Theory, Lecture Notes in Mathematics, vol. 1104, Springer, Berlin, 1984, pp. 217–232.CrossRefGoogle Scholar
Jensen, D. and Ehrenfeucht, A., Some problem in elementary arithmetics . Fundamenta Mathematicae, vol. 92 (1976), pp. 223–245.CrossRefGoogle Scholar
Kaye, R., Models of Peano Arithmetic, Oxford Logic Guides, vol. 15, Oxford Science Publications, New York, 1991.CrossRefGoogle Scholar
Krajíček, J. and Pudlák, P., On the structure of initial segments of models of arithmetic . Archive for Mathematical Logic, vol. 28 (1988), no. 2, pp. 91–98.CrossRefGoogle Scholar
Lindström, P., Aspects of Incompleteness, Lecture Notes in Logic, vol. 10, Springer-Verlag, Berlin, 1997.CrossRefGoogle Scholar
McAloon, K., On the complexity of models of arithmetic, this Journal, vol. 47 (1982), no. 2, pp. 403–415.Google Scholar
Misercque, D., Branches of the E-tree which are not isomorphic , Bulletin of the London Mathematical Society, vol. 17 (1985), pp. 513–517.Google Scholar
Mostowski, A., A generalization of the incompleteness theorem . Fundamenta Mathematicae, vol. 49 (1961), pp. 205–232.Google Scholar
Simmons, H., Large discrete parts of the E-tree, this Journal, vol. 53 (1988), no. 3, pp. 980–984.Google Scholar
Smoryński, C., The incompleteness theorems , Handbook of Mathematical Logic (Barwise, J., editor), North-Holland, Amsterdam, 1977, pp. 821–865.CrossRefGoogle Scholar
Smoryński, C., Fifty years of self-reference in arithmetic . Notre Dame Journal of Formal Logic, vol. 22 (1981), no. 4, pp. 357–378.Google Scholar
Solovay, R. M., Injecting inconsistencies into models of PA . Annals of Pure and Applied Logic, vol. 44 (1989), pp. 101–132.Google Scholar