u that do not have a corresponding element in the model. The model, however, is consistent, because the absence of these elements cannot be observed through first-order logic. With u as the reals, these missing elements would correspond to undefinable numbers. Skolem used the term "relative" to describe this state of affairs, where the same set is included in two models of set theory, is countable in one model and not countable in the other model. He described this as the "most important" result in his paper. Contemporary set theorists describe concepts that do not depend on the choice of a transitive model as absolute. From their point of view, Skolem's paradox simply shows that countability is not an absolute property in first-order logic. (Kunen 1980 p. 141; Enderton 2001 p. 152; Burgess 1977 p. 406). Skolem described his work as a critique of (first-order) set theory, intended to illustrate its weakness as a foundational system: "I believed that it was so clear that axiomatisation in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come for a critique." (Ebbinghaus and van Dalen, 2000, p. 147) Reception by the mathematical community A central goal of early research into set theory was to find a first-order axiomatisation for set theory which was categorical, meaning that the axioms would have exactly one model, consisting of all sets. Skolem's result showed that this is not possible, creating doubts about the use of set theory as a foundation of mathematics. It took some time for the theory of first-order logic to be developed enough for mathematicians to understand the cause of Skolem's result; no resolution of the paradox was widely accepted during the 1920s. Fraenkel (1928) still described the result as an antinomy: "Neither have the books yet been closed on the antinomy, nor has agreement on its significance and possible solution yet been reached." (van Dalen and Ebbinghaus, 2000, p. 147). In 1925, von Neumann presented a novel axiomatisation of set theory, which developed into NBG set theory. Very much aware of Skolem's 1922 paper, von Neumann investigated countable models of