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Church's thesis, continuity, and set theory

Published online by Cambridge University Press:  12 March 2014

M. Beeson  and
A. Ščedrov
M. Beeson*
Affiliation:
Rijksuniversiteit Utrecht
A. Ščedrov
Affiliation:
Utrecht, State University of New YorkBuffalo, New York 14214
*
Department of Mathematics and Computer Science, San Jose State University, San Jose, California 95192
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Abstract

Under the assumption that all “rules” are recursive (ECT) the statement Cont(NN, N) that all functions from NN to N are continuous becomes equivalent to a statement KLS in the language of arithmetic about “effective operations”. Our main result is that KLS is underivable in intuitionistic Zermelo-Fraenkel set theory + ECT. Similar results apply for functions from R to R and from 2N to N. Such results were known for weaker theories, e.g. HA and HAS. We extend not only the theorem but the method, fp-realizability, to intuitionistic ZF.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 49 , Issue 2 , June 1984 , pp. 630 - 643
Copyright
Copyright © Association for Symbolic Logic 1984

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References

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