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Gödel introduced the original provability predicate in the proofs of Gödel’s incompleteness theorems, and Rosser defined a new one. They are equivalent in the standard model ${\mathbb N}$ of arithmetic or any nonstandard model of ${\rm PA} + {\rm Con_{PA}} $ , but the behavior of Rosser’s provability predicate is different from the original one in nonstandard models of ${\rm PA} + \neg {\rm Con_{PA}} $ . In this paper, we investigate several properties of the derivability conditions for Rosser provability predicates, and prove the existence of a Rosser provability predicate with which we can define any consistent complete extension of ${\rm PA}$ in some nonstandard model of ${\rm PA} + \neg {\rm Con_{PA}} $ . We call it a universal Rosser predicate. It follows from the theorem that the true arithmetic ${\rm TA}$ can be defined as the set of theorems of ${\rm PA}$ in terms of a universal Rosser predicate in some nonstandard model of ${\rm PA} + \neg {\rm Con_{PA}} $ . By using this theorem, we also give a new proof of a theorem that there is a nonstandard model M of ${\rm PA} + \neg {\rm Con_{PA}} $ such that if N is an initial segment of M which is a model of ${\rm PA} + {\rm Con_{PA}} $ then every theorem of ${\rm PA}$ in N is a theorem of $\rm PA$ in ${\mathbb N}$ . In addition, we prove that there is a Rosser provability predicate such that the set of theorems of $\rm PA$ in terms of the Rosser provability predicate is inconsistent in any nonstandard model of ${\rm PA} + \neg {\rm Con_{PA}} $ .

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[1] Arai, T., Derivability conditions on Rosser’s provability predicates . Notre Dame Journal of Formal Logic, vol. 31 (1990), no. 4, pp. 487497.
[2] Guaspari, D. and Solovay, R. M., Rosser sentences . Annals of Mathematical Logic, vol. 16 (1979), no. 1, pp. 8199.
[3] Hájek, P. and Pudlák, P., Two orderings of the class of all countable models of Peano arithmetic , Model Theory of Algebra and Arithmetic, Lecture Notes in Mathematics, vol. 834, Springer, Berlin Heidelberg, 1980, pp. 174185.
[4] Hájek, P. and Pudlák, P., Metamathematics of First-Order Arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1993.
[5] Kikuchi, M. and Kurahashi, T., Illusory models of Peano arithmetic, this Journal, vol. 81 (2016), pp. 11631175.
[6] Kreisel, G. and Takeuti, G., Formally self-referential propositions for cut free classical analysis and related systems . Dissertationes Mathematicae (Rozprawy Matematyczne), vol. 118 (1974).
[7] Kurahashi, T., Henkin sentences and local reflection principles for Rosser provability . Annals of Pure and Applied Logic, vol. 167 (2016), no. 2, pp. 7394.
[8] Lindström, P., Aspects of Incompleteness, second ed., Lecture Notes in Logic, vol. 10, Peters, A K, Natick, MA, 2003.
[9] Mostowski, A., A generalization of the incompleteness theorem . Fundamenta Mathematicae, vol. 49 (1961), pp. 205232.
[10] Rosser, J. B., Extensions of some theorems of Gödel and Church, this Journal, vol. 1 (1936), no. 3, pp. 8791.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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