Expressing infinity without foundation

The axiom of infinity can be expressed by stating the existence of sets satisfying a formula which involves restricted universal quantifiers only, even if the axiom of foundation is not assumed. The problem of expressing the existence of infinite sets in the first order settheoretic language by means of formulae of low logical complexity has been addressed in [PP88] and [PP9Ob]. While the usual formulations of the infinity axiom (Inf) make use of formulae involving (at least) alternations of universal and existential restricted quantifiers, [PP88] provided the first example of a formula involving only restricted universal quantifiers, whose satisfiability entails the existence of infinite sets, provided the foundation axiom (FA) is assumed together with the usual axioms of Zermelo-Fraenkel except, of course, the infinity axiom. It was then observed in [PP9Ob] that an even shorter formula had the same property. As explained in [PP88], the above problem is related to the so-called decision problem for fragments of set theory (see [CFO90]). Set theories not assuming FA but rather contradicting it in various forms have come to attract considerable interest (see [Acz88]), and the corresponding decision problem has begun to be investigated (see [PP9Oa]). It is therefore of particular interest to ask whether there are restricted purely universal formulae which are satisfiable but not finitely satisfiable, even when FA is dropped. In this note we show that a positive answer can be obtained through an appropriate merging of the two formulae in [PP88] and [PP9Ob], although neither of them suffices alone. Let SE be the first order set-theoretic language with identity, based on the membership relation e. A formula of Ad is restricted if it does not contain quantifiers except for the restricted quantifiers (Vx e y) and (3x e y). Let ZFdenote ZF Inf and ZF-denote ZF FA. In ZF-one can define the ordinals as transitive sets well-ordered by e and the nonzero natural numbers as successor ordinals with zero and successor ordinals only, as elements. Finiteness is taken to stand for equinumerousity with a natural number, and Inf can be stated as the existence of a set containing all the natural numbers. Received May 30, 1990; revised November 14, 1990. This work was supported by funds from the MPI, and by the AXL project of ENI and ENIDATA. ? 1991, Association for Symbolic Logic 0022-4812/91/5604-0005/$01 .60