axiomatisation is satisfied by any infinite structure, then the same axioms are satisfied by some countable structure. In particular, this implies that if the first-order versions of Zermelo's axioms of set theory are satisfiable, they are satisfiable in some countable model. The same is true for any consistent first-order axiomatisation of set theory. The paradoxical result and its mathematical implications Skolem (1922) pointed out the seeming contradiction between the Löwenheim–Skolem theorem on the one hand, which implies that there is a countable model of Zermelo's axioms, and Cantor's theorem on the other hand, which states that uncountable sets exist, and which is provable from Zermelo's axioms. "So far as I know," Skolem writes, "no one has called attention to this peculiar and apparently paradoxical state of affairs. By virtue of the axioms we can prove the existence of higher cardinalities... How can it be, then, that the entire domain B [a countable model of Zermelo's axioms] can already be enumerated by means of the finite positive integers?" (Skolem 1922, p. 295, translation by Bauer-Mengelberg). More specifically, let B be a countable model of Zermelo's axioms. Then there is some set u in B such that B satisfies the first-order formula saying that u is uncountable. For example, u could be taken as the set of real numbers in B. Now, because B is countable, there are only countably many elements c such that c ∈ u according to B, because there are only countably many elements c in B to begin with. Thus it appears that u should be countable. This is Skolem's paradox. Skolem went on to explain why there was no contradiction. In the context of a specific model of set theory, the term "set" does not refer to an arbitrary set, but only to a set that is actually included in the model. The definition of countability requires that a certain one-to-one correspondence, which is itself a set, must exist. Thus it is possible to recognise that a particular set u is countable, but not countable in a particular model of set theory, because there is no set in the model that gives a one-to-one correspondence between u and the natural numbers in that model. From an interpretation of the model into our conventional notions of these sets, this means that although u maps to an uncountable set, there are many elements in our intuitive notion of