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

What has been the historical relationship between set theory and logic? On the óne 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 question 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.