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A note on Mathematics of infinity

  • Erik Palmgren (a1)

In the paper Mathematics of infinity, Martin-Löf extends his intuitionistic type theory with fixed “choice sequences”. The simplest, and most important instance, is given by adding the axioms

to the type of natural numbers. Martin-Löf's type theory can be regarded as an extension of Heyting arithmetic (HA). In this note we state and prove Martin-Löf's main result for this choice sequence, in the simpler setting of HA and other arithmetical theories based on intuitionistic logic (Theorem A). We also record some remarkable properties of the resulting systems; in general, these lack the disjunction property and may or may not have the explicit definability property. Moreover, they represent all recursive functions by terms.

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[1]Kreisel, G., Mathematical significance of consistency proofs, this Journal, vol. 23 (1958), pp. 155182.
[2]Laugwitz, D., Ω-calculus as a generalization of field extension—An alternative approach to nonstandard analysis, Nonstandard analysis—Recent developments (Hurd, A. E., editor), Lecture Notes in Mathematics, vol. 983, Springer-Verlag, Berlin and New York, 1983.
[3]Liu, S.-C., A proof-theoretic approach to nonstandard analysis with emphasis on distinguishing between constructive and nonconstructive results, The Kleene symposium (Barwise, J., Keisler, H. J., and Kunen, K., editors), North-Holland, Amsterdam, 1980, pp. 391414.
[4]Martin-Löf, P., Mathematics of infinity, COLOG-88 computer logic (Martin-Löf, P. and Mints, G. E., editors), Lecture Notes in Computer Science, vol. 417, Springer-Verlag, Berlin and New York, 1989.
[5]Mycielski, J., Analysis without actual infinity, this Journal, vol. 46 (1981), pp. 625633.
[6]Troelstra, A. S. (editor), Metamathematical investigation of intuitionistic arithmetic and analysis, Lecture Notes in Mathematics, vol. 344, Springer-Verlag, Berlin and New York, 1973.
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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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