Functional Interpretations of Constructive Set Theory in All Finite Types

Dialectica 62 (2):149-177 (2008)
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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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From the weak to the strong existence property.Michael Rathjen - 2012 - Annals of Pure and Applied Logic 163 (10):1400-1418.

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References found in this work

Constructive set theory.John Myhill - 1975 - Journal of Symbolic Logic 40 (3):347-382.
On the interpretation of intuitionistic number theory.S. C. Kleene - 1945 - Journal of Symbolic Logic 10 (4):109-124.

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