Hume's Principle requires the existence of the finite cardinals and their cardinal, but these are the only cardinals the Principle requires. Were the Principle an analysis of the concept of cardinal number, it would already be peculiar that it requires the existence of any cardinals; an analysis of bachelor is not expected to yield unmarried men. But that it requires the existence of some cardinals, the countable ones, but not others, the uncountable, makes it seem invidious; it is as if an analysis of people required that there be men but not women, or whites but not blacks. If we deprive the Principle of existential commitments, it will cease to yield Dedekind's axioms for the natural numbers and so fail a good test of material adequacy. But since there are cardinals no second-order theory guarantees, neither can the Principle be beefed up to require all cardinals.
"Skolem Redux." Notre Dame J. Formal Logic 41 (4) 399 - 414, 2000. https://doi.org/10.1305/ndjfl/1038336883