Zermelo s1937

Abstract

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.