Church's thesis, continuity, and set theory

Journal of Symbolic Logic 49 (2):630-643 (1984)
Under the assumption that all "rules" are recursive (ECT) the statement $\operatorname{Cont}(N^N,N)$ that all functions from N N 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 2 N 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
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Andrew W. Swan (2014). CZF Does Not Have the Existence Property. Annals of Pure and Applied Logic 165 (5):1115-1147.
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