Short Proofs for Slow Consistency

Abstract

Let $Con(\mathbf{T})\upharpoonright x$ denote the finite consistency statement “there are no proofs of contradiction in $\mathbf{T}$ with $\le x$ symbols.” For a large class of natural theories $\mathbf{T}$, Pudlák has shown that the lengths of the shortest proofs of $Con(\mathbf{T})\upharpoonright n$ in the theory $\mathbf{T}$ itself are bounded by a polynomial in $n$. At the same time he conjectures that $\mathbf{T}$ does not have polynomial proofs of the finite consistency statements $Con(\mathbf{T}+Con(\mathbf{T}))\upharpoonright n$. In contrast, we show that Peano arithmetic ($\mathbf{PA}$) has polynomial proofs of $Con(\mathbf{PA}+{Con}^{\ast }(\mathbf{PA}))\upharpoonright n$, where ${Con}^{\ast }(\mathbf{PA})$ 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 $Con(\mathbf{PA})$ is equivalent to ${\epsilon }_0$ iterations of slow consistency. Our argument is proof-theoretic, whereas previous investigations of slow consistency relied on nonstandard models of arithmetic.