Functional interpretations of constructive set theory in all finite types

Dialectica 62 (2):149–177 (2008)
Gödel's dialectica interpretation of Heyting arithmetic HA may be seen as expressing a lack of confidence in our understanding of unbounded quantification. Instead of formally proving an implication with an existential consequent or with a universal antecedent, the dialectica interpretation asks, under suitable conditions, for explicit 'interpreting' instances that make the implication valid. For proofs in constructive set theory CZF-, it may not always be possible to find just one such instance, but it must suffice to explicitly name a set consisting of such interpreting instances. The aim of eliminating unbounded quantification in favor of appropriate constructive functionals will still be obtained, as our ∧-interpretation theorem for constructive set theory in all finite types CZFω- shows. By changing to a hybrid interpretation ∧q, we show closure of CZFω- under rules that – in stronger forms – have already been studied in the context of Heyting arithmetic. In a similar spirit, we briefly survey modified realizability of CZFω- and its hybrids. Central results of this paper have been proved by Burr 2000a and Schulte 2006, however, for different translations. We use a simplified interpretation that goes back to Diller and Nahm 1974. A novel element is a lemma on absorption of bounds which is essential for the smooth operation of our translation.
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DOI 10.1111/j.1746-8361.2008.01133.x
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References found in this work BETA
Joseph R. Shoenfield (1967). Mathematical Logic. Reading, Mass.,Addison-Wesley Pub. Co..
A. S. Troelstra (1988). Constructivism in Mathematics: An Introduction. Sole Distributors for the U.S.A. And Canada, Elsevier Science Pub. Co..
John Myhill (1975). Constructive Set Theory. Journal of Symbolic Logic 40 (3):347-382.

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Michael Rathjen (2012). From the Weak to the Strong Existence Property. Annals of Pure and Applied Logic 163 (10):1400-1418.

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