 In 1915, Löwenheim published his landmark “Über Möglichkeiten im Relativkalkül”. This paper, written in the tradition of the Peirce-Schroeder calculus of relatives, established the first significant metalogical theorem; from certain points of view, it marks the beginning of model theory. Löwenheim considered a class of what he called “counting expressions” (Zählausdrücke) whose quantifiers range only over the domain of objects in the universe, but not over relatives; he then proved that, for any such counting expression, if it is satisfiable, it is satisfiable in some denumerable domain. In modern terminology, his “counting expressions” are formulas of first-order logic; but his terminology shows no influence either from Peirce’s logic of “first intention”, or from Russell’s theory of types. Löwenheim, like all logicians of this era, did not possess the distinction between an object language and a metalanguage. His proof is difficult to follow, and the precise details of his theorem—of what he believed he had proved, and what he had, in fact, proved—have been the subject of extensive scholarly discussion. (A survey of the differing interpretations is provided by Mancosu, Zach, &amp; Badesa 2009, and a detailed reconstruction of the proof itself by Badesa 2004.) The paper appears to have had no influence until Skolem sharpened and extended its results in his 1920. Löwenheim, like Peirce and Russell, did not isolate an axiomatic system encompassing first-order logic, nor did he draw a distinction between syntax and semantics. Still less does he argue that his class of “counting expressions” is in some way logically privileged, and provides a favored foundation for mathematics. The Löwenheim theorem was in time to be recognized as isolating a fundamental property of first-order logic. But the full implications of his result were not to become clear until later, after Hilbert had introduced the metamathematical study of logical systems. (Incidentally, Löwenheim credited the elegant \(\Sigma\) and \(\Pi\) symbolism of Peirce for suggesting the infinitary expansions that were necessary to his proof; and it is difficult to see how he could have obtained his theorem with any of the other quantificational notations then on offer. He was still vigorously defending the advantages of the Peirce-Schroeder notation against the notation of Principia as late as Löwenheim 1940.)