Arithmetic as a Theory Modulo

Abstract

We present constructive arithmetic in Deduction modulo with rewrite rules only. In natural deduction and in sequent calculus, the cut elimination theorem and the analysis of the structure of cut free proofs is the key to many results about predicate logic with no axioms: analyticity and non-provability results, completeness results for proof search algorithms, decidability results for fragments, constructivity results for the intuitionistic case. . . Unfortunately, the properties of cut free proofs do not extend in the presence of axioms and the cut elimination theorem is not as powerful in this case as it is in pure logic. This motivates the extension of the notion of cut for various axiomatic theories such as arithmetic, Church’s simple type theory, set theory and others. In general, we can say that a new axiom will necessitate a specific extension of the notion of cut: there still is no notion of cut general enough to be applied to any axiomatic theory. Deduction modulo [2, 3] is one attempt, among others, towards this aim. In deduction modulo, a theory is not a set of axioms but a set of axioms combined with a set of rewrite rules. For instance, the axiom ∀x x + 0 = x can be replaced by the rewrite rule x + 0 −→ x. The point is that replacing the axiom by the rewrite rule introduces short-cuts in the corresponding proofs, which avoid axiomatic cuts. When the set of rewrite rules is empty, one is simply back to regular predicate logic. On the other hand, when the set of axioms is empty we have theories expressed by rewrite rules only. For such theories, cut free proofs are similar to cut free proofs in pure logic, in particular they end with an introduction rule. Thus, when a theory can be expressed in deduction modulo with rewrite rules only and, in addition, cuts can be eliminated modulo these rewrite rules, the theory has most of the properties of pure logic. This leads to the question of which theories can be expressed with rewrite rules only in such a way that cut-elimination holds. It is known that several theories can be expressed in such a setting, for instance all equational theories, type theory, set theory, etc. . . But arithmetic was an important example of a theory that lacked such a presentation. The goal of this paper is to show that arithmetic can indeed be presented in deduction modulo without axioms in such a way that cut elimination holds. The cut elimination result is built using the generic tools introduced in [3]. When considering arithmetic, it is customary to keep the cut-elimination argument predicative. We show that these generic tools also make it possible to build a predicative proof. It should be noticed that second-order arithmetic can be embedded in simple type theory with the axiom of infinity and thus that it can be expressed in deduction modulo. Our presentation of first-order arithmetic in deduction modulo uses many ideas coming from second-order arithmetic. However, our presentation of arithmetic has exactly the power of first-order arithmetic.