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Intensional interpretations of functionals of finite type I

  • W. W. Tait (a1)
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T0 will denote Gödel's theory T[3] of functionals of finite type (f.t.) with intuitionistic quantification over each f.t. added. T1 will denote T0 together with definition by bar recursion of type o, the axiom schema of bar induction, and the schema

of choice. Precise descriptions of these systems are given below in §4. The main results of this paper are interpretations of T0 in intuitionistic arithmetic U0 and of T1 in intuitionistic analysis is U1. U1 is U0 with quantification over functionals of type (0,0) and the axiom schemata AC00 and of bar induction.

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[1]Brouwer, L. E. J., Über Definitionsbereiche von Funktionen, Mathematische Annalen, vol. 97 (1927), pp. 6076.
[2]Gödel, K., Zur intuitionistischen Arithmetik und Zahlentheorie, Ergebnisse eines math. Koll., Heft. 4 für 19311932 (1933), pp. 3438.
[3]Göodel, K., Über eine bisher noch nicht benützte Erweiterung des finiteti Standpunktes, Dialectica, vol. 12 (1958), pp. 280287.
[4]Grzegorczyk, A., Recursive objects in all finite types, Fundamenta Mathematicae, vol. 54 (1964), pp. 7393.
[5]Kleene, S. C., Countable functionals, Constructivity in mathematics, North Holland Publishing Co., Amsterdam, 1959, pp. 81100.
[6]Kleene, S. C., Recursive functionals and quantifiers of finite type I, Transactions of the American Mathematical Society, vol. 91 (1959), pp. 156.
[7]Kreisel, G., Interpretation of classical analysis by means of constructive functionals of finite type, Constructivity in mathematics, North Holland Publishing Co., Amsterdam, 1959, pp. 101128.
[8]Kreisel, G., Inessential extensions of Heyting's arithmetic by means of functionals of finite type, this Journal, vol. 24, no. 3 (1959), p. 284 (Abstract).
[9]Spector, C., Provably recursive functionals of analysis: A consistency proof of analysis by an extension of principles formulated in current intuitionistic mathematics, Recursive function theory, Proceedings of symposia in pure mathematics, American Mathematical Society, Providence, R.I., 1962, pp. 127.
[10]Tait, W. W., A second order theory of functionals of higher type (to appear).
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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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