Reduction games, provability, and compactness

Abstract

Hirschfeldt and Jockusch (2016) introduced a two-player game in which winning strategies for one or the other player precisely correspond to implications and non-implications between $\Pi^1_2$ principles over $\omega$-models of $\mathsf{RCA}_0$. They also introduced a version of this game that similarly captures provability over $\mathsf{RCA}_0$. We generalize and extend this game-theoretic framework to other formal systems, and establish a certain compactness result that shows that if an implication $\mathsf{Q} \to \mathsf{P}$ between two principles holds, then there exists a winning strategy that achieves victory in a number of moves bounded by a number independent of the specific run of the game. This compactness result generalizes an old proof-theoretic fact noted by H.~Wang (1981), and has applications to the reverse mathematics of combinatorial principles. We also demonstrate how this framework leads to a new kind of analysis of the logical strength of mathematical problems that refines both that of reverse mathematics and that of computability-theoretic notions such as Weihrauch reducibility, allowing for a kind of fine-structural comparison between $\Pi^1_2$ principles that has both computability-theoretic and proof-theoretic aspects, and can help us distinguish between these, for example by showing that a certain use of a principle in a proof is "purely proof-theoretic", as opposed to relying on its computability-theoretic strength. We give examples of this analysis to a number of principles at the level of $\mathsf{B}\Sigma^0_2$, uncovering new differences between their logical strengths.