 Skolem’s first major papers were his 1920 and especially his 1922. In the first he proved (or re-proved) in a more perspicuous form the downward Löwenheim-Skolem theorem. In the second he provided a new proof of that result. He also criticized Zermelo’s axiom of separation, which had taken the form: Given a set S and a definite proposition \(\phi(x)\), there exists a set S’ of all elements s of S such that \(\phi(s)\). Here the concept of “definite proposition” was left somewhat imprecise. Skolem’s proposal was to identify “definite propositions” with the formulas of first-order logic (with identity). Although Skolem declared this identification to be “natural” and “completely clear”, he did not explicitly argue for the restriction of quantifiers to the first level. He then gave the earliest satisfactory first-order formulation of Zermelo’s set theory, and then applied the Löwenheim-Skolem result to obtain the Skolem paradox. 