  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.
