system
Given the following post, respond as if you're commenting on Reddit:
human
Why do mathematicians prefer first-order logic over second-order logic?

I know that FOL has more deductive power and SOL has more expressive power. I'm a bit confused as to why deductive power is preferred over expressive power. Shouldn't isomorphism be valued? Is the historical trend towards favoring FOL and deductive power just a matter of taste/a historical accident?
gpt
Mathematicians, for the most part, don't really prefer first-order logic over second-order logic. The vast majority of working mathematicians do their work in a system unique to the sub-field they're working in. The reason the formalization, par excellence, for all these different fields is first-order set theory is a historical accident. In 1908, Zermelo published his axioms for set theory, and there's reason to think he conceived of it as second-order. But it had a couple of key flaws, and in 1922 when Fraenkel reformulated the system, he did so as a first-order system. It was this formulation that gained traction, and so it was the first-order formulation that became standard. Additionally, people were fairly skeptical of second-order systems, associating them with the failure of Frege's logicist project in the Grundgesetze.