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 ${\mathrm{\Pi }}_2^1$ principles over $\omega$-models of ${RCA}_0$. They also introduced a version of this game that similarly captures provability over ${RCA}_0$. We generalize and extend this game-theoretic framework to other formal systems, and establish a certain compactness result that shows that if an implication $Q\to P$ 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 ${\mathrm{\Pi }}_2^1$ 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 $B{\mathrm{\Sigma }}_2^0$, uncovering new differences between their logical strengths.

Hirschfeldt 和 Jockusch (2016) 介绍了一种双人游戏，其中一个或另一个玩家的获胜策略恰好对应于${\mathrm{\pi }}_{\mathrm{2个}}^{\mathrm{1个}}$原则超过$\omega$-模型${RC\mathrm{一个}}_0$. 他们还介绍了该游戏的一个版本，该版本同样捕获了可证明性${RC\mathrm{一个}}_0$. 我们将这个博弈论框架推广并扩展到其他形式系统，并建立一定的紧凑性结果，表明如果蕴涵$问\to P$在两个原则之间成立，则存在一种获胜策略，该策略通过独立于特定游戏运行的数字限制的多个移动来取得胜利。这个紧致性结果推广了 H. Wang (1981) 指出的一个古老的证明理论事实，并应用于组合原理的逆​​向数学。我们还展示了这个框架如何导致对数学问题逻辑强度的一种新分析，它改进了逆向数学和可计算性理论概念（例如 Weihrauch 可还原性）的逻辑强度，允许在两者之间进行一种精细结构比较${\mathrm{\pi }}_{\mathrm{2个}}^{\mathrm{1个}}$具有可计算性理论和证明理论方面的原理，并且可以帮助我们区分它们，例如通过证明在证明中对原理的某种使用是“纯粹的证明理论”，而不是依赖于它的可计算性-理论强度。我们给出了这种分析的例子，以在水平上的一些原则$乙{\mathrm{\Sigma }}_{\mathrm{2个}}^0$，发现他们的逻辑优势之间的新差异。