Is a logicist bound to the claim that as a matter of analytic truth there is an actual infinity of objects? If Hume’s Principle is analytic then in the standard setting the answer appears to be yes. Hodes’s work pointed to a way out by offering a modal picture in which only a potential infinity was posited. However, this project was abandoned due to apparent failures of cross-world predication. We re-explore this idea and discover that in the setting of the potential infinite one can interpret first-order Peano arithmetic, but not second-order Peano arithmetic. We conclude that in order for the logicist to weaken the metaphysically loaded claim of necessary actual infinities, they must also weaken the mathematics they recover.

逻辑学家是否必须坚持这样的主张，即作为分析真理的问题，存在实际无限的对象？如果休谟原理是分析性的，那么在标准设置中答案似乎是肯定的。Hodes 的工作通过提供模态图指出了一条出路，其中只假定了一个潜在的无穷大。然而，由于跨世界预测的明显失败，这个项目被放弃了。我们重新探索这个想法，发现在势无穷大的情况下，可以解释一阶皮亚诺算术，但不能解释二阶皮亚诺算术。我们得出结论，为了让逻辑学家削弱必然实际无穷大的形而上学主张，他们也必须削弱他们恢复的数学。