countability". Hunter (1971, p. 208) describes the contradiction as "hardly even a paradox". Fraenkel et al. (1973, p. 304) explain that contemporary mathematicians are no more bothered by the lack of categoricity of first-order theories than they are bothered by the conclusion of Gödel's incompleteness theorem that no consistent, effective, and sufficiently strong set of first-order axioms is complete. Countable models of ZF have become common tools in the study of set theory. Forcing, for example, is often explained in terms of countable models. The fact that these countable models of ZF still satisfy the theorem that there are uncountable sets is not considered a pathology; van Heijenoort (1967) describes it as "a novel and unexpected feature of formal systems" (van Heijenoort 1967, p. 290). See also References Stephen Cole Kleene, (1952, 1971 with emendations, 1991 10th printing), Introduction to Metamathematics, North-Holland Publishing Company, Amsterdam NY. . cf pages 420-432: § 75. Axiom systems, Skolem's paradox, the natural number sequence. Stephen Cole Kleene, (1967). Mathematical Logic. English translation: External links Vaughan Pratt's celebration of his academic ancestor Skolem's 120th birthday Extract from Moore's discussion of the paradox(link to archive, original is broken link) Inner model theory Mathematical paradoxes Model theory de:Löwenheim-Skolem-Theorem#Das Skolem-Paradoxon