Beyond first-order logic: the historical interplay between mathematical logic and axiomatic set theory

History and Philosophy of Logic 1 (1-2):95-137 (1980)
Abstract
What has been the historical relationship between set theory and logic? On the one hand, Zermelo and other mathematicians developed set theory as a Hilbert-style axiomatic system. On the other hand, set theory influenced logic by suggesting to Schröder, Löwenheim and others the use of infinitely long expressions. The questions of which logic was appropriate for set theory - first-order logic, second-order logic, or an infinitary logic - culminated in a vigorous exchange between Zermelo and Gödel around 1930
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Ignacio Jané (1993). A Critical Appraisal of Second-Order Logic. History and Philosophy of Logic 14 (1):67-86.

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