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A CONSTRUCTIVE EXAMINATION OF A RUSSELL-STYLE RAMIFIED TYPE THEORY

Published online by Cambridge University Press:  26 April 2018

ERIK PALMGREN
ERIK PALMGREN*
Affiliation:
DEPARTMENT OF MATHEMATICS STOCKHOLM UNIVERSITY 106 91 STOCKHOLM, SWEDEN E-mail: palmgren@math.su.se
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Abstract

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.

Keywords

03B1503F3503F50ramified type theoryintuitionistic logicreducibility axiom
Type
Communications
Information
Bulletin of Symbolic Logic , Volume 24 , Issue 1 , March 2018 , pp. 90 - 106
Copyright
Copyright © The Association for Symbolic Logic 2018 

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References

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