Abstract
This paper describes an axiomatic theory BT, which is a suitable formal theory for developing constructive mathematics, due to its expressive language with countable number of set types and its constructive properties such as the existence and disjunction properties, and consistency with the formal Church thesis. BT has a predicative comprehension axiom and usual combinatorial operations. BT has intuitionistic logic and is consistent with classical logic. BT is mutually interpretable with a so called theory of arithmetical truth PATr and with a second-order arithmetic SA that contains infinitely many sorts of sets of natural numbers. We compare BT with some standard second-order arithmetics and investigate the proof-theoretical strengths of fragments of BT, PATr and SA.
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References
Beeson, M., Some relations between classical and constructive mathematics, The Journal of Symbolic Logic 43(2):228–246, 1978.
Beeson, M., A type-free Gödel interpretation, The Journal of Symbolic Logic, 43(2):213–227, 1978.
Beeson, M., Foundations of Constructive Mathematics. Metamathematical Studies, Springer, Berlin, 1985.
Bishop, E., Foundations of Constructive Analysis, Ishi Press International, New York, 2012.
Bishop, E., and D. Bridges, Constructive Analysis, Springer, Heidelberg, 2011.
Dragalin, A. G., Mathematical Intuitionism. Introduction to Proof Theory, American Mathematical Society, Providence, 1987.
Feferman, S., A language and axioms for explicit mathematics, Lecture Notes in Mathematics 450:87–139, 1975.
Feferman, S., Constructive theories of functions and classes, Logic Colloquium 78 (Mons) 450:159–224, 1979.
Friedman, H., The consistency of classical set theory relative to a set theory with intuitionistic logic, The Journal of Symbolic Logic 38(2):315–319, 1973.
Friedman, H., Subsystems of set theory and analysis, PhD thesis, Massachusetts Institute of Technology, 1967.
Gordeev, L., Proof-theoretic analysis: weak systems of functions and classes, Annals of Pure and Applied Logic, 38(1):1–121, 1988.
Kachapova, F., Interpretation of constructive multi-typed theory in the theory of arithmetical truth, Lobachevskii Journal of Mathematics 36(4):332–341, 2015.
Kachapova, F., Realizability and existence property of a constructive set theory with types, Proceedings of the 13th Asian Logic Conference 2015, pp. 136–155.
Kashapova, F., Isolation of classes of constructively derivable theorems in a many-sorted intuitionistic set theory equivalent to a second-order arithmetic, Doklady Academii Nauk 29(3):583–587, 1984.
Mendelson, E., Introduction to Mathematical Logic, Chapman and Hall/CRC, Boca Raton, 2009.
Simpson, S.G., Subsystems of Second Order Arithmetic, Cambridge University Press, Cambridge, 2010.
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Kachapova, F. Metamathematical Properties of a Constructive Multi-typed Theory. Stud Logica 105, 587–610 (2017). https://doi.org/10.1007/s11225-016-9701-0
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DOI: https://doi.org/10.1007/s11225-016-9701-0