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In this article we examine the natural interpretation of a ramified type hierarchy into Martin-Löf type theory with an infinite sequence of universes. It is shown that under this predicative interpretation some useful special cases of Russell’s reducibility axiom are valid, namely functional reducibility. This is sufficient to make the type hierarchy usable for development of constructive mathematical analysis in the style of Bishop. We present a ramified type theory suitable for this purpose. One may regard the results of this article as an alternative solution to the problem of the proliferation of levels of real numbers in Russell’s theory, which avoids impredicativity, but instead imposes constructive logic. The intuitionistic ramified type theory introduced here also suggests that there is a natural associated notion of predicative elementary topos.

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[1]Aczel, P., The type-theoretic interpretation of constructive set theory, Logic Colloquium ’77 (Macintyre, A., Pacholski, L., and Paris, J., editors), North-Holland, Amsterdam, 1978, pp. 5566.
[2]Bishop, E., Foundations of Constructive Analysis, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1967.
[3]Bell, J. L., Toposes and Local Set Theories, Oxford University Press, New York, 1988.
[4]Feferman, S., Iterated inductive fixed-point theories: Application to Hancock’s conjecture, Patras Logic Symposium (Metakides, G., editor), North-Holland, Amsterdam, 1982, pp. 171196.
[5]Kamareddine, F., Laan, T., and Nederpelt, R., Types in logic and mathematics before 1940, this Bulletin, vol. 8 (2002), pp. 185–245.
[6]Laan, T. and Nederpelt, R., A modern elaboration of the Ramified theory of types. Studia Logica, vol. 57 (1996), pp. 243278.
[7]Martin-Löf, P., Intuitionistic Type Theory, Notes by Giovanni Sambin of a series of lectures given in Padova 1980, Bibliopolis, Naples, 1984.
[8]Martin-Löf, P., An intuitionistic theory of types, Twenty-Five Years of Constructive Type Theory (Smith, J. M. and Sambin, G., editors), Oxford University Press, New York, 1998, pp. 127177.
[9]Myhill, J., A refutation of an unjustified attack on the axiom of reducibility, Bertrand Russell Memorial Volume (Russell, B. and Roberts, G. W., editors), Routledge, New York, 1979, pp. 8190.
[10]Palmgren, E., Intuitionistic Ramified type theory, Oberwolfach Reports, vol. 5, no. 2, European Mathematical Society, Zurich, 2008, pp. 943946.
[11]Ramsey, F. P., The foundations of mathematics. Proceedings of the London Mathematical Society, Series 2, vol. 24 (1926), pp. 338384.
[12]Russell, B., Mathematical logic as based on the theory of types. American Journal of Mathematics, vol. 30 (1908), pp. 222262.
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Bulletin of Symbolic Logic
  • ISSN: 1079-8986
  • EISSN: 1943-5894
  • URL: /core/journals/bulletin-of-symbolic-logic
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