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- W4287692978 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." @default.
- W4287692978 created "2022-07-26" @default.
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- W4287692978 date "2020-08-03" @default.
- W4287692978 modified "2023-09-29" @default.
- W4287692978 title "Reduction games, provability, and compactness" @default.
- W4287692978 doi "https://doi.org/10.48550/arxiv.2008.00907" @default.
- W4287692978 hasPublicationYear "2020" @default.
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