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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.

03F40 Gödel numberings and issues of incompleteness
03B45 Modal logic (including the logic of norms)
03F30 First-order arithmetic and fragments
Full Text: DOI
[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
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[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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