Slim Models of Zermelo Set Theory

Working in Z+KP , we give a new proof that the class of hereditarily finite sets cannot be proved to be a set in Zermelo set theory, extend the method to establish other failures of replacement, and exhibit a formula Φ(λ, a) such that for any sequence 〈Aλ | λ a limit ordinal 〉 where for each λ, Aλ ⊆ 2, there is a supertransitive inner model of Zermelo containing all ordinals in which for every λ Aλ = {a | Φ(λ, a)}. Preliminaries This paper explores the weakness of Zermelo set theory, Z, as a vehicle for recursive definitions. We work in the system Z +KP , which adds to the axioms of Zermelo those of Kripke–Platek set theory KP . Z + KP is of course a subsystem of the familiar system ZF of Zermelo–Fraenkel. Mention is made of the axiom of choice, but our constructions do not rely on that Axiom. It is known that Z+KP +AC is consistent relative to Z: see [M2], to appear as [M3], which describes inter alia a method of extending models of Z + AC to models of Z + AC + KP . We begin by reviewing the axioms of the two systems Z and KP . In the formulation of KP , we shall use the familiar Lévy classification of formulæ: ∆0 formulæ are those in which every quantifier is restricted, ∀x(x ∈ y =⇒ . . .) or ∃x(x ∈ y & . . .), which we write as ∀x :∈y . . . and ∃x :∈y . . . respectively. In all such cases x and y must be distinct variables. Π1 and Σ1 formulæ are those respectively of the form ∀xB and ∃xB, where B is a ∆0 formula. 0·0 The axioms of the system Z are Extensionality ∀z(z ∈ x ⇐⇒ z ∈ y) =⇒ x = y, Empty Set ∅ ∈ V , Pairing {x, y} ∈ V , Union ⋃ x ∈ V , Power Set P(x) ∈ V , Foundation ∀x [x 6= ∅ =⇒ ∃y :∈x (x ∩ y = ∅)], Infinity ω ∈ V , and for each class A the axiom