Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-27T23:30:48.763Z Has data issue: false hasContentIssue false
  • English
  • Français

A STRONG MULTI-TYPED INTUITIONISTIC THEORY OF FUNCTIONALS

Published online by Cambridge University Press:  22 July 2015

FARIDA KACHAPOVA
FARIDA KACHAPOVA*
Affiliation:
SCHOOL OF COMPUTER AND MATHEMATICAL SCIENCES AUCKLAND UNIVERSITY OF TECHNOLOGY AUCKLAND, NEW ZEALANDE-mail: farida.kachapova@aut.ac.nz
Get access Rights & Permissions [Opens in a new window]

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.

Keywords

03F5503F50Intuitionistic theorylawless sequencecreating subjectKripke schemaBeth modelforcingconsistencytruth predicate
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 3 , September 2015 , pp. 1035 - 1065
Copyright
Copyright © The Association for Symbolic Logic 2015 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Bernini, S., A very strong intuitionistic theory. Studia Logica, vol. 35 (1976), no. 4, pp. 377–385.CrossRefGoogle Scholar
Bernini, S., A note on my paper “A very strong intuitionistic theory”.Studia Logica, vol. 37 (1978), no. 4, pp. 349–350.CrossRefGoogle Scholar
Brouwer, L.E.J., Essentially negative properties, Collected works. Philosophy and foundations of mathematics (Heyting, A., editor), vol. 1, Amsterdam, 1975, pp. 478–479.Google Scholar
Dragalin, A. G., Mathematical intuitionism. Introduction to proof theory, American Mathematical Society, Providence, RI, 1987.Google Scholar
Friedman, H., Set theoretic foundations for constructive analysis. Annals of Mathematics (2), vol. 105 (1977), no. 2, pp. 1–28.CrossRefGoogle Scholar
Kachapova, F., A generalization of Beth model to functionals of high types, Proceedings of the 12th Asian Logic Conference, (2013), pp. 185–209.CrossRefGoogle Scholar
Kachapova, F., A strong intuitionistic theory of functionals, http://arxiv.org/abs/1403.2813, 2014.Google Scholar
Kashapova, F., Intuitionistic theory of functionals of higher type.Mathematical Notes, vol. 45 (1989), no. 3, pp. 66–79.CrossRefGoogle Scholar
Kolmogorov, A. N. and Dragalin, A. G., Introduction to mathematical logic, Moscow University, Moscow, 1982.Google Scholar
Lyubetskii, V. A., Transfer theorems and intuitionistic set theory.Doklady Akademii Nauk, vol. 357 (1997), no. 2, pp. 168–171.Google Scholar
McNaughton, R., Some formal relative consistency proofs, this Journal, vol. 18 (1953), no. 2, pp. 136–144.Google Scholar
Mendelson, E., Introduction to mathematical logic, Chapman and Hall/CRC, Boca Raton, Florida, 2009.Google Scholar
Myhill, J., Formal systems of intuitionistic analysis. Logic, Methodology and Philos. Sci. III, (1968), pp. 161–178.Google Scholar
Tarski, A., The concept of truth in formalized languages. Studia Philosophica, vol. 1(1935), pp. 261–405.Google Scholar
van Dalen, D, An interpretation of intuitionistic analysis. Annals of Mathematical Logic, vol. 13 (1978), pp. 1–43.CrossRefGoogle Scholar
Wendel, N., The inconsistency of Bernini’s very strong intuitionistic theory. Studia Logica, vol. 37 (1978), no. 4, pp. 341–347.CrossRefGoogle Scholar