Lifschitz realizability for intuitionistic Zermelo–Fraenkel set theory

Abstract

A variant of realizability for Heyting arithmetic which validates Church's thesis with uniqueness condition, but not the general form of Church's thesis, was introduced by V. Lifschitz in [15]. A Lifschitz counterpart to Kleene's realizability for functions (in Baire space) was developed by van Oosten [19]. In that paper he also extended Lifschitz' realizability to second order arithmetic. The objective here is to extend it to full intuitionistic Zermelo-Fraenkel set theory, IZF. The machinery would also work for extensions of IZF with large set axioms. In addition to separating Church's thesis with uniqueness condition from its general form in intuitionistic set theory, we also obtain several interesting corollaries. The interpretation repudiates a weak form of countable choice, AC ω,ω , asserting that a countable family of inhabited sets of natural numbers has a choice function. AC ω,ω is validated by ordinary Kleene realizability and is of course provable in ZF. On the other hand, a pivotal consequence of AC ω,ω , namely that the sets of Cauchy reals and Dedekind reals are isomorphic, remains valid in this interpretation.Another interesting aspect of this realizability is that it validates the lesser limited principle of omniscience.MSC:03E25,03E35,03E70,03F25, 03F35,03F50,03F55,03F60

Separating Fragments of Wlem, Lpo, and Mp

“…In fact, any (non-trivial) Kripke model over a linear order with no last element and which contains the standard natural numbers (such as the full model over the partial order) satisfies both WLEM and LPO and falsifies LEM. LLPO and WKL are separated in Lifschitz realizability [3]. Kleene's number realizability K 1 separates MP and LLPO ω [15], giving a whole column of non-reversals; we give a very different model of the same separation below (Theorem 12).…”

Indefiniteness in Semi-Intuitionistic Set Theories: On a Conjecture of Feferman

“…There are several essentially different forms of realizability for set theories to choose from (cf. [4,5,14,18,19,23,22,24,25,26,28]). Moreover, what should the realizers be and how should the realizability universe be defined?…”

“…. 5 Since V 0 satisfies AC there is an injective function F in V 0 with domain R V 0 whose range is a set of ordinals. Identifying…”

“…Note that there are sets A, A and hereditarily finite sets e, e such that [e] L[A] (R L[A] ) A CH as well as [e ] L[A ] (R L[A ] )A ¬CH, and a fortiori there exists a set A such that [e ] L[A] (R L[A]) A CH ∨ ¬CH for some hereditarily finite e 5. We cannot choose C to beR V 0 since L[R V 0 ] = L (cf.…”

Constructive Zermelo-Fraenkel Set Theory, Power Set, and the Calculus of Constructions

“…Realizability based on indices of general set recursive functions was introduced in [33] and employed to prove, inter alia, metamathematical properties for CZF augmented by strong forms of the axiom of choice in [34,Theorems 8.3,8.4]. There are points of contact with a notion of realizability used by Tharp [39] who employed (indices of) Σ 1 definable partial (class) functions as realizers, though there are important differences, too, as Tharp works in a classical context and assumes a definable search operation on the universe which basically amounts to working under the hypothesis V = L. Moreover, there are connections with Lifschitz' realizability [16] where a realizer for an existential arithmetical statement provides a finite co-recursive set of witnesses (see [25,9] for extensions to analysis and set theory). We adopt the conventions and notations from the previous section.…”