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On the intuitionistic strength of monotone inductive definitions

  • Sergei Tupailo (a1)

We prove here that the intuitionistic theory T0↾ + UMIDN. or even EETJ↾ + UMIDN, of Explicit Mathematics has the strength of –CA0. In Section 1 we give a double-negation translation for the classical second-order μ-calculus, which was shown in [Mö02] to have the strength of –CA0. In Section 2 we interpret the intuitionistic μ-calculus in the theory EETJ↾ + UMIDN. The question about the strength of monotone inductive definitions in T0 was asked by S. Feferman in 1982, and — assuming classical logic — was addressed by M. Rathjen.

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[Fef82] S. Feferman , Monotone inductive definitions, The L. E. J. Brouwer centenary symposium ( A. S. Troelstra and D. van Dallen , editors), North-Holland, Amsterdam, 1982, pp. 7789.

[GRS97] T. Glaß , M. Rathjen , and A. Schlüter , On the proof-theoretic strength of monotone induction in explicit mathematics, Annals of Pure and Applied Logic, vol. 85 (1997), pp. 146.

[Tak89] S. Takahashi , Monotone inductive definitions in a constructive theory of functions and classes, Annals of Pure and Applied Logic, vol. 42 (1989), pp. 255297.

[Tu03] S. Tupailo , Realization of constructive set theory into Explicit Mathematics: a lower bound for impredicative Mahlo universe, Annals of Pure and Applied Logic, vol. 120 (2003), no. 1–3, pp. 165196.

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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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