We analyze the precise modal commitments of several natural varieties of set-theoretic potentialism, using tools we develop for a general model-theoretic account of potentialism, building on those of Hamkins, Leibman and Löwe [14], including the use of buttons, switches, dials and ratchets. Among the potentialist conceptions we consider are: rank potentialism (true in all larger $V_\beta $ ), Grothendieck–Zermelo potentialism (true in all larger $V_\kappa $ for inaccessible cardinals $\kappa $ ), transitive-set potentialism (true in all larger transitive sets), forcing potentialism (true in all forcing extensions), countable-transitive-model potentialism (true in all larger countable transitive models of ZFC), countable-model potentialism (true in all larger countable models of ZFC), and others. In each case, we identify lower bounds for the modal validities, which are generally either S4.2 or S4.3, and an upper bound of S5, proving in each case that these bounds are optimal. The validity of S5 in a world is a potentialist maximality principle, an interesting set-theoretic principle of its own. The results can be viewed as providing an analysis of the modal commitments of the various set-theoretic multiverse conceptions corresponding to each potentialist account.

我们分析了几种自然变体的集合论势能论的精确模态承诺，使用我们为势能论的一般模型理论解释开发的工具，建立在 Hamkins、Leibman 和 Löwe [14] 的基础上，包括按钮、开关的使用、刻度盘和棘轮。我们考虑的势能论概念包括：秩势势论（在所有较大的 $V_\beta $ 中为真）、格洛腾迪克-策梅罗势势论（在所有较大的 $V_\kappa $ 中对于不可接近的基数 $\kappa $ 都为真） )、传递集潜在主义（在所有较大的传递集中都为真）、强制潜在主义（在所有强制扩展中都为真）、可数传递模型潜在主义（在 ZFC 的所有较大可数传递模型中为真）、可数模型潜在主义（在ZFC 的所有较大的可数模型）等。在每种情况下，我们确定模态有效性的下限，通常是 S4.2 或 S4.3，以及 S5 的上限，证明在每种情况下这些界限都是最优的。S5 在一个世界中的有效性是一个潜在的最大化原则，它本身就是一个有趣的集合论原则。结果可以被视为提供了对与每个潜在主义帐户相对应的各种集合论多元宇宙概念的模态承诺的分析。