Home

# ILLUSORY MODELS OF PEANO ARITHMETIC

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.

References
Hide All
[1] Craig, W., On axiomatizability within a system, this Journal, vol. 18 (1953), pp. 3032.
[2] Guaspari, D. and Solovay, R. M., Rosser sentences . Annals of Mathematical Logic, vol. 16 (1979), no. 1, pp. 8199.
[3] Hájek, P., On a new notion of partial conservativity , Computation and Proof Theory, Lecture Notes in Mathematics, vol. 1104, Springer, Berlin, 1984, pp. 217232.
[4] Jensen, D. and Ehrenfeucht, A., Some problem in elementary arithmetics . Fundamenta Mathematicae, vol. 92 (1976), pp. 223245.
[5] Kaye, R., Models of Peano Arithmetic, Oxford Logic Guides, vol. 15, Oxford Science Publications, New York, 1991.
[6] 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. 9198.
[7] Lindström, P., Aspects of Incompleteness, Lecture Notes in Logic, vol. 10, Springer-Verlag, Berlin, 1997.
[8] McAloon, K., On the complexity of models of arithmetic, this Journal, vol. 47 (1982), no. 2, pp. 403415.
[9] Misercque, D., Branches of the E-tree which are not isomorphic , Bulletin of the London Mathematical Society, vol. 17 (1985), pp. 513517.
[10] Mostowski, A., A generalization of the incompleteness theorem . Fundamenta Mathematicae, vol. 49 (1961), pp. 205232.
[11] Simmons, H., Large discrete parts of the E-tree, this Journal, vol. 53 (1988), no. 3, pp. 980984.
[12] Smoryński, C., The incompleteness theorems , Handbook of Mathematical Logic (Barwise, J., editor), North-Holland, Amsterdam, 1977, pp. 821865.
[13] Smoryński, C., Fifty years of self-reference in arithmetic . Notre Dame Journal of Formal Logic, vol. 22 (1981), no. 4, pp. 357378.
[14] Solovay, R. M., Injecting inconsistencies into models of PA . Annals of Pure and Applied Logic, vol. 44 (1989), pp. 101132.
Recommend this journal

The Journal of Symbolic Logic
• ISSN: 0022-4812
• EISSN: 1943-5886
• URL: /core/journals/journal-of-symbolic-logic
Who would you like to send this to? *

×

## Full text viewsFull text views reflects the number of PDF downloads, PDFs sent to Google Drive, Dropbox and Kindle and HTML full text views.

Total number of HTML views: 0
Total number of PDF views: 59 *