Cut elimination for Zermelo set theory

We show how to express intuitionistic Zermelo set theory in deduction modulo (i.e. by replacing its axioms by rewrite rules) in such a way that the corresponding notion of proof enjoys the normalization property. To do so, we first rephrase set theory as a theory of pointed graphs (following a paradigm due to P. Aczel) by interpreting set-theoretic equality as bisimilarity, and show that in this setting, Zermelo’s axioms can be decomposed into graph-theoretic primitives that can be turned into rewrite rules. We then show that the theory we obtain in deduction modulo is a conservative extension of (a minor extension of) Zermelo set theory. Finally, we prove the normalization of the intuitionistic fragment of the theory. The cut elimination theorem is a central result in proof theory that has many corollaries such as the disjunction property and the witness property for constructive proofs, the completeness of various proof search methods and the decidability of some fragments of predicate logic, as well as some independence results. However, most of these corollaries hold for pure predicate logic and do not generally extend when we add axioms, because the property that cut-free proofs end with an introduction rule does not generalize in the presence of axioms. Thus, extensions of the normalization theorem have been proved for some axiomatic theories, for instance arithmetic, simple type theory [11, 12] or the socalled stratified foundations [4]. There are several ways to extend normalization to axiomatic theories: the first is to consider a special form of cut corresponding to a given axiom, typically the induction axiom. A second is to transform axioms into deduction rules, typically the β-equivalence axiom. A third way is to replace axioms by computation rules and consider deduction rules modulo the congruence generated by these computation rules [5, 7]. Unfortunately, extending the normalization theorem to set theory has always appeared to be difficult or even impossible: a counter example, due to M. Crabbé [3] shows that normalization does not hold when we replace the axioms of set theory by the obvious deduction rules, and in particular the Restricted Comprehension axiom by a deduction rule allowing to deduce the formula a ∈ b ∧ P (x ← a) from a ∈ {x ∈ b | P} and vice-versa. In the same way,