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Transfinite induction and bar induction of types zero and one, and the role of continuity in intuitionistic analysis

Abstract

The following is a self-contained proof theoretic treatment of two of the principal axiom schemata of current intuitionistic analysis: the axiom of bar induction (Brouwer's bar theorem) and the axiom of continuity. The results are formulated in terms of formal derivability in elementary intuitionistic analysis H(§ 1), so the positive (i.e., derivability) results also apply to elementary classical analysis Z1 (Appendix 1). Both schemata contain the combination of quantifiers νfΛn, where f, g, … are intended to range over free choice sequences of suitable kinds of objects x, y, …; for example, natural numbers or sequences of natural numbers, and n, m, p, r, … over natural numbers (non-negative integers).

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[1] L. E.J. Brouwer , Über Definitionsbereiche von Funktionen, Mathematische Annalen, vol. 97 (1927), pp. 6075.

[2] L. E. J. Brouwer , Points and spaces, Canadian Journal of Mathematics, vol. 6 (1954), pp. 117.

[5] K. Gödel , Über eine bisher noch nicht benützte Erweiterung des finiten Standpunktes, Dialectica, vol. 12 (1958), pp. 280287.

[8] S. C. Kleene , Arithmetical predicates and function quantifiers, Transactions of the American Mathematical Society, vol. 79 (1955), pp. 312340.

[10] S. C. Kleene and R. E. Vesley , Foundations of Intuitionistic Mathematics, North-Holland Publishing Co., Amsterdam, 1965.

[16] C. Spector , Provably recursive functionals of analysis: a consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics, Proceedings of the Symposia in Pure Mathematics, vol. 5, American Mathematical Society, Provindence, 1962, pp. 127.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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