From the weak to the strong existence property

A hallmark of many an intuitionistic theory is the existence property, EP, i.e., if the theory proves an existential statement then there is a provably definable witness for it. However, there are well known exceptions, for example, the full intuitionistic Zermelo-Fraenkel set theory, IZF, does not have the existence property, where IZF is formulated with Collection. By contrast, the version of intuitionistic Zermelo-Fraenkel set theory formulated with Replacement, IZFR, has the existence property. Moreover, IZF does not even enjoy a weaker form of the existence property, wEP, defined by the slackened requirement of finding a provably definable set of witnesses for every existential theorem. In view of these results, one might be tempted to put the blame for the failure of the existence properties squarely on Collection. However, in this paper it is shown that several well known intuitionistic set theories with Collection have the weak existence property. Among these theories are CZF, CZFE , and CZFP , i.e., respectively, constructive Zermelo-Fraenkel set theory (CZF) without subset collection, CZF formulated with Exponentiation and also CZF augmented by the Power Set axiom (basically IZF with only bounded separation). As a result, the culprit preventing the weak existence property from obtaining must consist of a combination of Collection and unbounded Separation. To bring about these results we introduce a form of realizability based on general set recursive functions where a realizer for an existential statement provides a set of witnesses for the existential quantifier rather than a single witness. Moreover, this notion of realizability needs to be combined with truth to yield the desired results. This form of realizability is also utilized, albeit shorn of its truth component, in showing partial conservativity results forCZF, CZFE , and CZFP over their intuitionistic counterparts IKP, IKP(E), and IKP(P), respectively. As it turns out, the combination of the weak existence property and partial conservativity of CZF over IKP plus a further ingredient can be used to show that CZF actually has the existence property. The additional ingredient is an advanced techniques from proof theory (cut elimination and ordinal analysis of IKP). Roughly the same techniques can be deployed in showing that CZFE and CZFP have the stronger existence property, too. However, this requires a new form of ordinal analysis for theories with Power Set and Exponentiation (cf. [39]) and is beyond the scope of the current paper. MSC:03F50, 03F35