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On the derivability of instantiation properties

Published online by Cambridge University Press:  12 March 2014

Harvey Friedman
Harvey Friedman*
Affiliation:
State University of New York at BuffaloAmherst, New York 14226
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Abstract

Every recursively enumerable extension of arithmetic which obeys the disjunction property obeys the numerical existence property [Fr, 1]. The requirement of recursive enumerability is essential. For extensions of intuitionistic second order arithmetic by means of sentences (in its language) with no existential set quantifiers, the numerical existence property implies the set existence property. The restriction on existential set quantifiers is essential. The numerical existence property cannot be eliminated, but in the case of finite extensions of HAS, can be replaced by a weaker form of it. As a consequence, the set existence property for intuitionistic second order arithmetic can be proved within itself.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 42 , Issue 4 , December 1977 , pp. 506 - 514
Copyright
Copyright © Association for Symbolic Logic 1977

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References

REFERENCES

[Fr, 1]Friedman, H., The disjunction property implies the numerical existence property, Proceedings of the National Academy of Sciences, vol. 72 (1975), no. 8, pp. 2877–2878.CrossRefGoogle ScholarPubMed
[Fr, 2]Friedman, H., Some applications of Kleene's methods for intuitionistic systems, Lecture Notes in Mathematics, vol. 337 (1973), pp. 113–170.CrossRefGoogle Scholar
[JS]de Jongh, D. H. J. and Smorynski, C., Kripke models and the theory of species, University of Amsterdam, Report 74–03, 1974.Google Scholar
[Kl, 1]Kleene, S. C., On the interpretation of intuitionistic number theory, this Journal, vol. 10 (1945), pp. 109–124.Google Scholar
[Kl, 2]Kleene, S. C., Disjunction and existence under implication in elementary intuitionistic formalisms, this Journal, vol. 27 (1962), pp. 11–18.Google Scholar
[KT]Kreisel, G. and Troelstra, A., Formal systems for some branches of intuitionistic analysis, Annals of Mathematical Logic, vol. 1 (1970), pp. 229–387.CrossRefGoogle Scholar
[My]Myhill, J., A note on indicator-functions, Proceedings of the American Mathematical Society, vol. 39 (1973), pp. 181–183.CrossRefGoogle Scholar
[Pr]Prawitz, D., Ideas and results in proof theory, Proceedings of the Second Scandinavian Logic Symposium (Fenstad, J. E., Editor), North-Holland, Amsterdam, 1971, pp. 235–307.CrossRefGoogle Scholar
[Sm]Smorynski, C., Applications of Kripke models, Lecture Notes in Mathematics, vol. 344 (1973), pp. 324–391.CrossRefGoogle Scholar
[Tr, 1]Troelstra, A., Notions of realizability for intuitionistic arithmetic and intuitionistic arithmetic in all finite types, Proceedings of the Second Scandinavian Logic Symposium (Fenstad, J. E., Editor), North-Holland, Amsterdam, 1971, pp. 369–405.CrossRefGoogle Scholar
[Tr, 2]Troelstra, A., Notes on intuitionistic second order arithmetic, Lecture Notes in Mathematics, vol. 337 (1973), pp. 171–205.CrossRefGoogle Scholar
[Tr, 3]Troelstra, A., Realizability and functional interpretations, Lecture Notes in Mathematics, vol. 344 (1973), pp. 175–274.CrossRefGoogle Scholar