Skip to main content Accessibility help
×
Home

A STRONG MULTI-TYPED INTUITIONISTIC THEORY OF FUNCTIONALS

  • FARIDA KACHAPOVA (a1)

Abstract

In this paper we describe an intuitionistic theory SLP. It is a relatively strong theory containing intuitionistic principles for functionals of many types, in particular, the theory of the “creating subject”, axioms for lawless functionals and some versions of choice axioms. We construct a Beth model for the language of intuitionistic functionals of high types and use it to prove the consistency of SLP.

We also prove that the intuitionistic theory SLP is equiconsistent with a classical theory TI. TI is a typed set theory, where the comprehension axiom for sets of type n is restricted to formulas with no parameters of types > n. We show that each fragment of SLP with types ≤ s is equiconsistent with the corresponding fragment of TI and that it is stronger than the previous fragment of SLP. Thus, both SLP and TI are much stronger than the second order arithmetic. By constructing the intuitionistic theory SLP and interpreting in it the classical set theoryTI, we contribute to the program of justifying classical mathematics from the intuitionistic point of view.

Copyright

References

Hide All
[1]Bernini, S., A very strong intuitionistic theory. Studia Logica, vol. 35 (1976), no. 4, pp. 377385.
[2]Bernini, S., A note on my paper “A very strong intuitionistic theory”.Studia Logica, vol. 37 (1978), no. 4, pp. 349350.
[3]Brouwer, L.E.J., Essentially negative properties, Collected works. Philosophy and foundations of mathematics (Heyting, A., editor), vol. 1, Amsterdam, 1975, pp. 478479.
[4]Dragalin, A. G., Mathematical intuitionism. Introduction to proof theory, American Mathematical Society, Providence, RI, 1987.
[5]Friedman, H., Set theoretic foundations for constructive analysis. Annals of Mathematics (2), vol. 105 (1977), no. 2, pp. 128.
[6]Kachapova, F., A generalization of Beth model to functionals of high types, Proceedings of the 12th Asian Logic Conference, (2013), pp. 185209.
[7]Kachapova, F., A strong intuitionistic theory of functionals, http://arxiv.org/abs/1403.2813, 2014.
[8]Kashapova, F., Intuitionistic theory of functionals of higher type.Mathematical Notes, vol. 45 (1989), no. 3, pp. 6679.
[9]Kolmogorov, A. N. and Dragalin, A. G., Introduction to mathematical logic, Moscow University, Moscow, 1982.
[10]Lyubetskii, V. A., Transfer theorems and intuitionistic set theory.Doklady Akademii Nauk, vol. 357 (1997), no. 2, pp. 168171.
[11]McNaughton, R., Some formal relative consistency proofs, this Journal, vol. 18 (1953), no. 2, pp. 136144.
[12]Mendelson, E., Introduction to mathematical logic, Chapman and Hall/CRC, Boca Raton, Florida, 2009.
[13]Myhill, J., Formal systems of intuitionistic analysis. Logic, Methodology and Philos. Sci. III, (1968), pp. 161178.
[14]Tarski, A., The concept of truth in formalized languages. Studia Philosophica, vol. 1(1935), pp. 261405.
[15]van Dalen, D, An interpretation of intuitionistic analysis. Annals of Mathematical Logic, vol. 13 (1978), pp. 143.
[16]Wendel, N., The inconsistency of Bernini’s very strong intuitionistic theory. Studia Logica, vol. 37 (1978), no. 4, pp. 341347.

Keywords

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed