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Rosser and Mostowski sentences. (English) Zbl 0688.03041
The authors provide a modal framework for the study of Rosser and Mostowski sentences of Peano Arithmetic. The properties of the operator called witness comparison, with which Rosser sentences can be discussed, have been analysed in a modal system by D. Guaspari and R. M. Solovay [Ann. Math. Logic 16, 81-99 (1979; Zbl 0426.03062)].
In the system, however, it has been shown that there are no fixed points for Rosser formulae. Several extensions of the system, therefore, are defined in this paper by adding new constants to fill the role of the fixed points so that not only Rosser sentences but also Mostowski sentences are discussed uniformly in a more complete modal setting. The main result is a lemma called intersection lemma, from which immediately follow the formalized Rosser theorem, the Friedman-Goldfarb-Harrington’s principle, and so on. The arithmetical completeness and semantical completeness by Kripke models for the systems are also proved.
Reviewer: O.Sonobe

03F30 First-order arithmetic and fragments
03B45 Modal logic (including the logic of norms)
Full Text: DOI
[1] Bernardi, C., Montagna, F.: Equivalence relations induced by extensional formulae. Fundam. Math.CXXIV, 221–233 (1984) · Zbl 0564.03043
[2] Boolos, G.: The unprovability of consistency. Cambridge: Cambridge University Press 1979 · Zbl 0409.03009
[3] Carlson, T.: Modal logics with several operators and provability interpretations · Zbl 0625.03007
[4] De Jongh, D.H.J.: A simplification of a completeness proof of Guaspari and Solovay · Zbl 0638.03018
[5] De Jongh, D.H.J., Montagna, F.: Generic generalized Rosser fixed points · Zbl 0638.03019
[6] Guaspari, D.: Sentences implying their own provability. J. Symb. Logic48, 777–789 (1983) · Zbl 0547.03035
[7] Guaspari, D., Solovay, R.: Rosser sentences. Ann. Math. Logic16, 81–99 (1979) · Zbl 0426.03062
[8] Montagna, F.: On the diagonalizable algebra of Peano Arithmetic. Boll. Unione Mat. Ital. V. Ser.,B 16, 795–812 (1979) · Zbl 0419.08010
[9] Mostowski, A.: A generalization of the incompleteness theorem. Fundam. Math.49, 205–232 (1961) · Zbl 0099.00604
[10] Rosser, J.B.: Extensions of some theorems of Gödel and Church. J. Symb. Logic 87–91 (1936) · Zbl 0015.33802
[11] Sambin, G.: An effective fixed point theorem in intuitionistic diagonalizable algebras. Stud. Logica35, 345–361 (1976) · Zbl 0357.02028
[12] Shepherdson, J.: Representability of recursively enumerable sets in formal systems. Arch. F. Math. Logik5, 119–127 (1960) · Zbl 0113.24305
[13] Smorynski, C.: Bimodal logic and arithmetic · Zbl 0503.03033
[14] Smorynski, C.: Self reference and modal logic. Berlin Heidelberg New York Tokyo: Springer 1985
[15] Smorynski, C.: Modal logic and self-reference. In Gabbay, D., Guenthner, F. (eds.) Handbook of philosophical logic II. Dordrecht: Reidel 1984
[16] Solovay, R.M.: Provability interpretations of modal logic. Isr. J. Math.25, 287–304 (1976) · Zbl 0352.02019
[17] Visser, A.: The provability logic of recursively enumerable theories extending Peano Arithmetic at arbitrary theories extending Peano Arithmetic. J. Philos. Logic13 (1984) · Zbl 0581.03009
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