his axioms in detail. In his concluding remarks, von Neumann comments that there is no categorical axiomatisation of set theory, or any other theory with an infinite model. Speaking of the impact of Skolem's paradox, he wrote: "At present we can do no more than note that we have one more reason here to entertain reservations about set theory and that for the time being no way of rehabilitating this theory is known." (Ebbinghaus and van Dalen, 2000, p. 148) Zermelo at first considered the Skolem paradox a hoax (van Dalen and Ebbinghaus, 2000, p. 148 ff.) and spoke against it starting in 1929. Skolem's result applies only to what is now called first-order logic, but Zermelo argued against the finitary metamathematics that underlie first-order logic (Kanamori 2004, p. 519 ff.). Zermelo argued that his axioms should instead be studied in second-order logic, a setting in which Skolem's result does not apply. Zermelo published a second-order axiomatisation in 1930 and proved several categoricity results in that context. Zermelo's further work on the foundations of set theory after Skolem's paper led to his discovery of the cumulative hierarchy and formalisation of infinitary logic (van Dalen and Ebbinghaus, 2000, note 11). Fraenkel et al. (1973, pp. 303–304) explain why Skolem's result was so surprising to set theorists in the 1920s. Gödel's completeness theorem and the compactness theorem were not proved until 1929. These theorems illuminated the way that first-order logic behaves and established its finitary nature, although Gödel's original proof of the completeness theorem was complicated. Leon Henkin's alternative proof of the completeness theorem, which is now a standard technique for constructing countable models of a consistent first-order theory, was not presented until 1947. Thus, in 1922, the particular properties of first-order logic that permit Skolem's paradox to go through were not yet understood. It is now known that Skolem's paradox is unique to first-order logic; if set theory is studied using higher-order logic with full semantics, then it does not have any countable models, due to the semantics being used. Current mathematical opinion Current mathematical logicians do not view Skolem's paradox as any sort of fatal flaw in set theory. Kleene (1967, p. 324) describes the result as "not a paradox in the sense of outright contradiction, but rather a kind of anomaly". After surveying Skolem's argument that the result is not contradictory, Kleene concludes: "there is no absolute notion of