Functional Interpretations of Constructive Set Theory in All Finite Types

Abstract

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.

From the weak to the strong existence property

“…also employs sets of witnesses to interpret existential quantifiers. It has been extended to set theories by Burr, Diller and Schulte[7,10,41]. Burr[7, Corollary 5.12] and Diller[10, Proposition 4.4] prove weak forms of term existence property for a higher type versions of CZF − .…”

“…It has been extended to set theories by Burr, Diller and Schulte[7,10,41]. Burr[7, Corollary 5.12] and Diller[10, Proposition 4.4] prove weak forms of term existence property for a higher type versions of CZF − . Interestingly, Diller conjectures ([10] Conjectures 4.8,4.9) that the term existence property fails for higher type versions of CZF − .…”

Functional Interpretations of Classical and Constructive Set Theory

“…For a proof, we refer to [9]. A weaker version of this lemma, adapted to the ×-translations, appears in [4] and [16] as 'distribution lemma'.…”

“…A detailed ∧-interpretation of CZF ω− is presented in [9]. We refer to this source for most of the proofs.…”

“…For the interpretation of laws governing the existential quantifier and of the schema of (Strong collection), we refer to [9].…”