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A Finite FirstOrder Theory of Classes
"... Abstract. We expose a formalism that allows the expression of any theory with one or more axiom schemes using a finite number of axioms. This allows us to give finite firstorder axiomatizations of arithmetic and real analysis, and a presentation of arithmetic in deduction modulo that has a finite n ..."
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Abstract. We expose a formalism that allows the expression of any theory with one or more axiom schemes using a finite number of axioms. This allows us to give finite firstorder axiomatizations of arithmetic and real analysis, and a presentation of arithmetic in deduction modulo that has a finite number of rewrite rules. Overall, this formalization relies on a weak calculus of explicit substitutions to provide a simple and finite framework. 1
A Semantic Proof that Reducibility Candidates entail Cut Elimination
, 2012
"... Two main lines have been adopted to prove the cut elimination theorem: the syntactic one, that studies the process of reducing cuts, and the semantic one, that consists in interpreting a sequent in some algebra and extracting from this interpretation a cutfree proof of this very sequent. A link bet ..."
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Two main lines have been adopted to prove the cut elimination theorem: the syntactic one, that studies the process of reducing cuts, and the semantic one, that consists in interpreting a sequent in some algebra and extracting from this interpretation a cutfree proof of this very sequent. A link between those two methods was exhibited by studying in a semantic way, syntactical tools that allow to prove (strong) normalization of proofterms, namely reducibility candidates. In the case of deduction modulo, a framework combining deduction and rewriting rules in which theories like Zermelo set theory and higher order logic can be expressed, this is obtained by constructing a reducibility candidates valued model. The existence of such a premodel for a theory entails strong normalization of its proofterms and, by the usual syntactic argument, the cut elimination property. In this paper, we strengthen this gate between syntactic and semantic methods, by providing a full semantic proof that the existence of a premodel entails the cut elimination property for the considered theory in deduction modulo. We first define a new simplified variant of reducibility candidates à la Girard, that is sufficient to prove weak normalization of proofterms (and therefore the cut elimination property). Then we build, from some model valued on the preHeyting algebra of those WN reducibility candidates, a regular model valued on a Heyting algebra on which we apply the usual soundness/strong completeness argument. Finally, we discuss further extensions of this new method towards normalization by evaluation techniques that commonly use Kripke semantics.