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Rosser orderings and free variables. (English) Zbl 0744.03058
Guaspari and Solovay have proposed the arithmetically complete modal system $$R$$ of provability in the case where Rosser sentences are taken into account. In this paper the authors prove that if free variables are included, the arithmetically complete system is not $$R$$ but $$R^ -$$ of Guaspari and Solovay, which has been introduced for the sake of $$R$$ first. From this completeness result, there follow the non-validity of some rules and that some principles concerning Rosser orderings cannot be decided even by the usual proof predicates in the case where free variables are included.

##### MSC:
 03F40 Gödel numberings and issues of incompleteness 03B45 Modal logic (including the logic of norms) 03F30 First-order arithmetic and fragments
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##### References:
 [1] Guaspari, D. and R.M. Solovay, Rosser Sentences, Annals of Mathematical Logic, 16, pp. 81-99, 1979. · Zbl 0426.03062 [2] de Jongh, D.H.J., Waiting for the Bell, in preparation, A. [3] de Jongh, D.H.J., A Simplification of a Completeness Proof of Guaspari and Solovay Studia Logica 46, pp. 187-192, 1987. · Zbl 0638.03018 [4] de Jongh, D.H.J. and F. Montagna, Provable Fixed Points, Zeitschrift f?r Mathematische Logik und Grundlagen der Mathematik 34, pp. 229-250, 1988. · Zbl 0661.03009 [5] Smory?ski, C., Self-reference and Modal Logic, Springer, New York, 1985. · Zbl 0596.03001 [6] Solovay, R.M., Provability Interpretations of Modal Logic, Israel Journal of Mathematics, vol. 25, pp. 287-304, 1976. · Zbl 0352.02019
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