 Zermelo's reply is dismissive, but there is something to the criticism. Certainly Zermelo's 1904 proof attempts to show that WOT can be proved while by-passing the general abstract theory of well-ordering and its association with the Cantorian ordinals, and therefore also bypassing the ‘the set W’ (as it was widely known) of all Cantorian ordinals (denoted ‘Ω’ by Cantor), and consequently the Burali-Forti antinomy. However, whatever Zermelo's intention, there is no explicit attempt to exclude the possibility that Lγ = W and thus the suggestion that antinomy might threaten. Of course, Zermelo, referring to critics who ‘base their objections upon the “Burali-Forti antinomy” ’, declares that this antinomy ‘is without significance for my point of view, since the principles I employed exclude the existence of a set W [of all ordinals]’ (Zermelo 1908a: 128, with earlier hints on 118–119) that the real problem is with the ‘more elementary’ Russell antinomy. It is also true that at the end of the 1904 paper, Zermelo states that the argument holds for those sets M ‘for which the totality of subsets, and so on, is meaningful’, which, in retrospect is clearly a hint at important restrictions on set formation. Even so, Zermelo's attitude is unfair. It could be that the remark about ‘the totality of subsets etc.’ is an indirect reference to difficulties with the comprehension principle, but even so the principle is not repudiated explicitly in the 1904 paper, neither does Zermelo put in its place another principle for the conversion of properties to sets, which is what the Aussonderungsaxiom of the 1908 axiomatisation does. Moreover, he does not say that the existence principles on which the proof is based are the only set existence principles, and he does not divorce the proof of the theorem from the Cantorian assumptions about well-ordering and ordinals. Indeed, Zermelo assumes that ‘every set can be well-ordered’ is equivalent to the Cantorian ‘every cardinality is an aleph’ (Zermelo 1904: 141). And despite his later claim (Zermelo 1908a: 119), he does appear to use the ordinals and the informal theory of well-ordering in his definition of γ-sets, where a γ-set is ‘any well-ordered Mγ…’, without any specification of how ‘well-ordered set’ is to be defined. What assurance is there that this can all be reduced to Zermelo's principles? One important point here is that it had not yet been shown that all the usual apparatus of set-theoretic mathematics (relations, ordering relations, functions, cardinal equivalence functions, order-isomorphisms, etc.) could be reduced to a few simple principles of set existence. All of this was to come in the wake of Zermelo's axiomatisation, and there is little doubt that this line of criticism greatly influenced the shape of the second proof given in 1908, of which a little more below. 