A Finite First-Order Presentation of Set Theory
Stéphane Vaillant, Institut National de Recherche en Informatique et en Automatique (INRIA), 78 - Rocquencourt (France). Unite de Recherche de Rocquencourt · OpenGrey (Institut de l'Information Scientifique et Technique) · 2001
We present a first-order formalization of set theory which has a finite number of axioms. Its syntax is similar to that often used in textbooks: it provides an encoding of the comprehension symbol. We prove that this formalization is a "conservative extension" of Zermelo's set theory. In fact the proof is more general and applies to other variants of Zermelo's set theory like ZF. This formalization rests upon an encoding of the comprehension binder in a language of explicit substitution. This presentation of set theory is also described as a deduction modulo system and the proof of equivalence is done within this formalism.