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Existence and feasibility in arithmetic

Published online by Cambridge University Press:  12 March 2014

Rohit Parikh
Rohit Parikh*
Affiliation:
Boston University, Boston, Massachusetts 02215
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Extract

“From two integers k, l one passes immediately to kl; this process leads in a few steps to numbers which are far larger than any occurring in experience, e.g., 67(257729).

Intuitionism, like ordinary mathematics, claims that this number can be represented by an arabic numeral. Could not one press further the criticism which intuitionism makes of existential assertions and raise the question: What does it mean to claim the existence of an arabic numeral for the foregoing number, since in practice we are not in a position to obtain it?

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 36 , Issue 3 , September 1971 , pp. 494 - 508
Copyright
Copyright © Association for Symbolic Logic 1972

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References

[B1]Bernays, P., Logical calculus, Institute for advanced study notes, Princeton, 1935–1936.Google Scholar
[B2]Bernays, P., On Platonism in Mathematics, Philosophy of mathematics, Prentice-Hall, New Jersey, 1964, pp. 274–88.Google Scholar
[BP]Benacerraf, P. and Putnam, H., Philosophy of mathematics, Prentice-Hall, 1964.Google Scholar
[E]Volpin, A. Esenine, The ultra-intuitionistic criticism and the antitraditional program for foundations of mathematics, Instuitionism and proof theory, Proceedings of the conference at Buffato 1968, North-Holland, Amsterdam, 1970.Google Scholar
[EM]Ehrenfeucht, A. and Mostowski, A., Models of theories admitting automorphisms, Fundamenta Mathematicae vol. 43 (1956), pp. 50–68.CrossRefGoogle Scholar
[G]Gödel, K., On the length of proofs, Undecidable, Davis, Martin, ed., Raven Press, 1965.Google Scholar
[HB]Hilbert, D., On the infinite, Philosophy of mathematics, Prentice-Hall, 1964, pp. 134–151.Google Scholar
[HB]Hilbert, D. and Bernays, P., Die Grundlagen der Mathematik, Springer, Vol. 1, 1936, Vol. 2, 1939.Google Scholar
[Kr]Kreisel, G., Hilbert's programme, Philosophy of mathematics, Prentice-Hall, 1964, pp. 157–182.Google Scholar
[Kr1]Kreisel, G., Axiomatisations of nonstandard analysis that are conservative extensions of formal systems for classical standard analysis, Applications of model theory, Holt, Rinehart and Winston, 1969.Google Scholar
[Ku]Kuroda, S., Classes of languages and linear bounded automata, Information and control, vol. 7 (1964), pp. 207–223.CrossRefGoogle Scholar
[L]Löb, M., Solution of a problem of Leon Henkin, this Journal, vol. 20 (1955), pp. 115–118.Google Scholar
[Pa]Parikh, R., A conservation result, Applications of model theory, Luxemburg, W. A. J., ed. Holt, Rinebart and Winston, New York, New York, 1969.Google Scholar
[Po]Poincar, H.é, The foundations of science, Science Press, 1913.Google Scholar
[R]Ritchie, R., Classes of predictably computable functions, Transactions of the American Mathematical Society, vol. 106 (1963), pp. 139–173.CrossRefGoogle Scholar
[S]Shepherdson, J., Nonstandard models for fragments of number theory, Symposium on the theory of models, North-Holland, Amsterdam, 1965.Google Scholar
[T]Tait, W., The e-substitutton method, this Journal, vol. 30 (1965), pp. 175–192.Google Scholar
[W]Wang, H., A survey of mathematical logic, North-Holland, Amsterdam, 1963.Google Scholar
[Gr]Grzegorczyk, A., Some classes of recursive functions, Rozprawy Matematyczne, vol. IV (1953), pp. 3–46.Google Scholar