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January 2020 Short Proofs for Slow Consistency
Anton Freund, Fedor Pakhomov
Notre Dame J. Formal Logic 61(1): 31-49 (January 2020). DOI: 10.1215/00294527-2019-0031


Let Con(T)x denote the finite consistency statement “there are no proofs of contradiction in T with x symbols.” For a large class of natural theories T, Pudlák has shown that the lengths of the shortest proofs of Con(T)n in the theory T itself are bounded by a polynomial in n. At the same time he conjectures that T does not have polynomial proofs of the finite consistency statements Con(T+Con(T))n. In contrast, we show that Peano arithmetic (PA) has polynomial proofs of Con(PA+Con(PA))n, where Con(PA) is the slow consistency statement for Peano arithmetic, introduced by S.-D. Friedman, Rathjen, and Weiermann. We also obtain a new proof of the result that the usual consistency statement Con(PA) is equivalent to ε0 iterations of slow consistency. Our argument is proof-theoretic, whereas previous investigations of slow consistency relied on nonstandard models of arithmetic.


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Anton Freund. Fedor Pakhomov. "Short Proofs for Slow Consistency." Notre Dame J. Formal Logic 61 (1) 31 - 49, January 2020.


Received: 5 May 2018; Accepted: 27 September 2018; Published: January 2020
First available in Project Euclid: 29 November 2019

zbMATH: 07196091
MathSciNet: MR4054244
Digital Object Identifier: 10.1215/00294527-2019-0031

Primary: 03F20
Secondary: 03F30 , 03F40

Keywords: finite consistency statements , polynomial proofs , slow consistency

Rights: Copyright © 2020 University of Notre Dame


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Vol.61 • No. 1 • January 2020
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