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Whither relevant arithmetic?

  • Harvey Friedman (a1) and Robert K. Meyer (a1)

Abstract

Based on the relevant logic R, the system R# was proposed as a relevant Peano arithmetic. R# has many nice properties: the most conspicuous theorems of classical Peano arithmetic PA are readily provable therein; it is readily and effectively shown to be nontrivial; it incorporates both intuitionist and classical proof methods. But it is shown here that R# is properly weaker than PA, in the sense that there is a strictly positive theorem QRF of PA which is unprovable in R#. The reason is interesting: if PA is slightly weakened to a subtheory P+, it admits the complex ring C as a model; thus QRF is chosen to be a theorem of PA but false in C. Inasmuch as all strictly positive theorems of R# are already theorems of P+, this nonconservativity result shows that QRF is also a nontheorem of R#. As a consequence, Ackermann's rule γ is inadmissible in R#. Accordingly, an extension of R# which retains its good features is desired. The system R##, got by adding an omega-rule, is such an extension. Central question: is there an effectively axiomatizable system intermediate between R# and R#, which does formalize arithmetic on relevant principles, but also admits γ in a natural way?

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References

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[1]Thistlewaite, P. B., McRobbie, M. A. and Meyer, R. K., Automated theorem-proving in non-classical logics, Pitman, London, 1988.
[2]Kripke, S. A., The problem of entailment, this Journal, vol. 24 (1959), p. 324. (Abstract)
[3]McRobbie, M. A., Meyer, R. K. and Thistlewaite, P. B., Towards efficient “knowledge-based” automated theorem proving for non-standard logics, Proceedings of the ninth international conference on automated deduction, Lecture Notes in Computer Science, vol. 310, Springer-Verlag, Berlin, 1988, pp. 197217.
[4]Meyer, R. K., Relevant arithmetic, Polish Academy of Sciences, Institute of Philosophy and Bulletin of the Section of Logic, vol. 5 (1976), pp. 133137.
[5]Kleene, S. C., Introduction to metamathematics, Van Nostrand, Princeton, New Jersey, 1952.
[6]Meyer, R. K., The consistency of arithmetic, typescript, 1975.
[7]Meyer, R. K. and Urbas, I., Conservative extension in relevant arithmetic, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 32 (1986), pp. 4550.
[8]Friedman, H., typescript, 1988.
[9]Tarski, A., A decision method for elementary algebra and geometry, 2nd ed., University of California Press, Berkeley, California, 1951.

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