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Polynomially and superexponentially shorter proofs in fragments of arithmetic

Published online by Cambridge University Press:  12 March 2014

Franco Montagna
Franco Montagna*
Affiliation:
Scuola di Specializzione in Logica Matematica, Università di Siena, 53100 Siena, Italy, E-mail: montagna@sivax.cineca.it
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Extract

In Parikh [71] it is shown that, if T is an r.e. consistent extension of Peano arithmetic PA, then, for each primitive recursive function g, there is a formula φ of PA such that

(In the following, Proof T(z, φ) and Prov T(φ) denote the metalinguistic assertions that z codes a proof of φ in T and that φ is provable in T respectively, where ProofT(z, ┌φ┐) and ProvT(┌φ┐) are the formalizations of Proof T(z,φ) and ProvT(φ) respectively in the language of PA, ┌φ┐ denotes the Gödel number of φ and ┌φ┐ denotes the corresponding numeral. Also, for typographical reasons, subscripts will not be made boldface.) If g is a rapidly increasing function, we express (1) by saying that ProvT(┌φ┐) has a much shorter proof modulo g than φ. Parikh's result is based on the fact that a suitable formula A(x), roughly asserting that (1) holds with x in place of φ, has only provable fixed points. In de Jongh and Montagna [89], this situation is generalized and investigated in a modal context. There, a characterization is given of arithmetical formulas arising from modal formulas of a suitable modal language which have only provable fixed points, and Parikh's result is obtained as a particular case.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 57 , Issue 3 , September 1992 , pp. 844 - 863
Copyright
Copyright © Association for Symbolic Logic 1992

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