 It must be noted that our formalization of naive set theory is an anachronism. Cantor did not fully formalize his set theory, so it cannot be determined whether his system falls afoul of the paradoxes (he did not think so, and there are some who agree with him now). Frege formalized his system more explicitly, but his system was not precisely a set theory in the modern sense: the most that can be said is that his system is inconsistent, for basically the reason given here, and a full account of the differences between Frege’s system and our “naive set theory” is beside the point (though historically certainly interesting).