On Skolemising Zermelo’s Set Theory

We give a Skolemised presentation of Zermelo’s set theory (with notations for comprehension, powerset, etc.) and show that this presentation is conservative w.r.t. the usual one (where sets are introduced by existential axioms). Conservativity is achieved by an explicit deskolemisation procedure that transforms terms and formulæ of the extended language into provably equivalent formulæ of the core language of set theory. Finally we show that the notation {t(x) | x ∈ u} (‘the set of all t(x) where x ranges over u’) is also definable in this framework, which proves that the weak form of replacement which is needed to define syntactic constructs such as (set-theoretic) λ-abstraction and infinitary Cartesian product does not need Fraenkel and Skolem’s replacement scheme to