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At the Scandinavian mathematics congress of 1922, Skolem drew an astonishing conclusion from a theorem on models of first-order theories that he had established in 1920. This theorem, now known as the Löwenheim-Skolem theorem, states that if a countable first-order theory has an infinite model at all, then it has a countable model. Löwenheim's earlier paper 1915 had handled the "one-sentence" case, and Skolem's generalization allowed arbitrary countable sets of sentences. Modulo Gödel's completeness theorem, the theorem is also expressed as "a consistent and countable first-order theory has a countable model". We will pass over the finer details of Skolem's proof and the formulations involved, and concentrate on the application made by Skolem in the context of set theory.

DOI: 10.1007/978-3-540-79384-7_36

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