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Might All Infinities Be the Same Size?

ABSTRACT Cantor proved that no set has a bijection between itself and its power set. This is widely taken to have shown that there infinitely many sizes of infinite sets. The argument depends on the principle that if two sets are of the same size, there is a bijection between them. I shall argue that it can be reasonable to be sceptical of this principle, and that there is an epistemic possibility that all infinite sets have the same size as the set of natural numbers and that there is even an epistemic possibility that the collection of all sets has the same size as the set of natural numbers. These epistemic possibilities depend on considerations about the epistemic possibility of the contingency of some pure sets.