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Cut elimination for Zermelo’s set theory
, 2006
"... 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 para ..."
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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 settheoretic equality as bisimilarity, and show that in this setting, Zermelo’s axioms can be decomposed into graphtheoretic 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 Stratified Foundations as a theory modulo
"... The Stratified Foundations are a restriction of naive set theory where the comprehension scheme is restricted to stratifiable propositions. It is known that this theory is consistent and that proofs strongly normalize in this theory. Deduction modulo is a formulation of firstorder logic with a ..."
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The Stratified Foundations are a restriction of naive set theory where the comprehension scheme is restricted to stratifiable propositions. It is known that this theory is consistent and that proofs strongly normalize in this theory. Deduction modulo is a formulation of firstorder logic with a general notion of cut. It is known that proofs normalize in a theory modulo if it has some kind of manyvalued model called a premodel. We show in this note that the Stratified Foundations can be presented in deduction modulo and that the method used in the original normalization proof can be adapted to construct a premodel for this theory. The Stratified Foundations are a restriction of naive set theory where the comprehension scheme is restricted to stratifiable propositions. This theory is consistent [8] and proofs in this theory strongly normalize [2], while naive set theory is contradictory and the consistency of the Stratified Foundations together with the extensionality axiom...