The second-order alternative in set theory

The paper discusses the natural second-order alternative to the 'weak' first-order formal systems of axiomatic set theory by focusing on the entangled relationship between second-order theories and first-order set theory. On the basis of a survey of some classical disputes on the problem, I argue that second-order logic is actually set theory in disguise, unless one has an independent notion of 'collection' or of 'property'. Moreover, I argue that if one does not have at one's disposal such independent notions then second-order logic may still be significant for set theory only if one presupposes that the powerset operation is completely determined independently of any axiomatization. I maintain that the philosophical price of both these options is too high.