This conclusion, that ascribing sizes to infinite sets should be ruled impossible, owing to the contradictory results obtained from these two ostensibly natural ways of attempting to do so, is a resolution to the problem that is consistent with, but less powerful than, the methods used in modern mathematics. The resolution to the problem may be generalized by considering Galileo's first definition of what it means for sets to have equal sizes, that is, the ability to put them in one-to-one correspondence. This turns out to yield a way of comparing the sizes of infinite sets that is free from contradictory results.