Realizability algebras II : new models of ZF + DC

The technology of classical realizability was developed in [15, 18] in order to extend the proof-program correspondence (also known as Curry-Howard correspondence) from pure intuitionistic logic to the whole of mathematical proofs, with excluded middle, axioms of ZF, dependent choice, existence of a well ordering on P (N), . . . We show here that this technology is also a new method in order to build models of ZF and to obtain relative consistency results. The main tools are : • The notion of realizability algebra [18], which comes from combinatory logic [2] and plays a role similar to a set of forcing conditions. The extension from intuitionistic to classical logic was made possible by Griffin’s discovery [7] of the relation between the law of Peirce and the instruction call-with-current-continuation of the programming language SCHEME. In this paper, we only use the simplest case of realizability algebra, which I call standard realizability algebra ; somewhat like the binary tree in the case of forcing. • The theory ZFε [13] which is a conservative extension of ZF, with a notion of strong membership, denoted as ε. The theory ZFε is essentially ZF without the extensionality axiom. We note an analogy with the Fraenkel-Mostowski models with “urelements” : we obtain a non well orderable set, which is a Boolean algebra denoted ג2, all elements of which (except 1) are empty. But we also notice two important differences : • The final model of ZF + ¬ AC is obtained directly, without taking a suitable submodel. • There exists an injection from the “pathological set” ג2 into R, and therefore R is also not well orderable. We show the consistency, relatively to the consistency of ZF, of the theory ZF + DC (dependent choice) with the following properties : there exists a sequence (Xn)n∈N of infinite subsets of R, the “cardinals” of which are strictly increasing (this means that there is an injection but no surjection from Xn to Xn+1), and such that Xm×Xn is equipotent with Xmn for m,n ≥ 2 ; there exists a sequence of infinite subsets of R, the “cardinals” of which are strictly decreasing.