Full Idea
The set-theory account of infinity doesn't just say that we can keep on counting, but that the natural numbers are an actual infinite set. This is necessary to make sense of the powerset of ω, as the set of all its subsets, and thus even bigger.
Gist of Idea
Set theory says that natural numbers are an actual infinity (to accommodate their powerset)
Source
James Robert Brown (Philosophy of Mathematics [1999], Ch. 5)
Book Reference
Brown,James Robert: 'Philosophy of Mathematics' [Routledge 2002], p.65
A Reaction
I don't personally find this to be sufficient reason to commit myself to the existence of actual infinities. In fact I have growing doubts about the whole role of set theory in philosophy of mathematics. Shows how much I know.