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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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T. Arai , Derivability conditions on Rosser’s provability predicates . Notre Dame Journal of Formal Logic, vol. 31 (1990), no. 4, pp. 487497.

D. Guaspari and R. M. Solovay , Rosser sentences . Annals of Mathematical Logic, vol. 16 (1979), no. 1, pp. 8199.

P. Hájek and P. Pudlák , 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.

P. Hájek and P. Pudlák , Metamathematics of First-Order Arithmetic, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1993.

T. Kurahashi , Henkin sentences and local reflection principles for Rosser provability . Annals of Pure and Applied Logic, vol. 167 (2016), no. 2, pp. 7394.

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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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