A Finite First-Order Presentation of Set Theory

Abstract

We present a first-order formalization of set theory which has a finite number of axioms. Its syntax is familiar since it provides an encoding of the comprehension symbol. Since this symbol binds a variable inon e of its arguments we let the givenformalizationrest upona calculus of explicit substitutionwith de Bruijnin dices. This presentationof set theory is also described as a deduction modulo system which is used as an intermediate system to prove that the given presentation is a conservative extension of Zermelo's set theory.