Zermelo accepted none of these approaches, for two reasons. First, he thought that the recursive definitions involved make direct use of the notion of finite number (a fact pointed out by Weyl 1910), which it ought to be the business of set theory to explain, not to presuppose. Secondly, he became aware that using essentially a first-order notion condemns the axiomatic system to countable models, the fundamental fact pointed out in Skolem (1923). His own approach was, first, to give a different kind of axiomatisation (see Zermelo 1929), and then to use (in Zermelo 1930) an essentially second-order notion in characterising the axiom of separation.[33]