On Skolemising Zermelo's Set Theory

Alexandre Miquel (1)

1 PPS - Preuves, Programmes et Systèmes

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 deskolemisationprocedure that transforms terms and formulae of the extended language into provably equivalent formulae of the core language of set theory. Finally we show that the notation {t(x) | x\in 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) lambda-abstraction and infinitary Cartesian product does not need Fraenkel and Skolem's replacement scheme to be justified.

Mots clés: Set theory; Skolem symbols; Replacement Scheme

Domaines: Logique en informatique [cs.LO]

Alexandre Miquel. On Skolemising Zermelo's Set Theory. 2008. ⟨hal-00425474⟩