Let denote the finite consistency statement “there are no proofs of contradiction in with symbols.” For a large class of natural theories , Pudlák has shown that the lengths of the shortest proofs of in the theory itself are bounded by a polynomial in . At the same time he conjectures that does not have polynomial proofs of the finite consistency statements . In contrast, we show that Peano arithmetic () has polynomial proofs of , where 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 is equivalent to iterations of slow consistency. Our argument is proof-theoretic, whereas previous investigations of slow consistency relied on nonstandard models of arithmetic.
"Short Proofs for Slow Consistency." Notre Dame J. Formal Logic 61 (1) 31 - 49, January 2020. https://doi.org/10.1215/00294527-2019-0031