Reduction Games, Provability, and Compactness

by Damir D. Dzhafarov, Denis R. Hirschfeldt, and Sarah C. Reitzes

Status: to appear in the Journal of Mathematical Logic

Availability: preprint

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 Π12 principles over ω-models of RCA0. They also introduced a version of this game that similarly captures provability over RCA0. We generalize and extend this game-theoretic framework to other formal systems, and establish a certain compactness result that shows that if an implication Q → 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 Π12 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 BΣ02, uncovering new differences between their logical strengths.