I recall once I read the short book "The Philosophy of Set Theory" [1] since I like philosophy and have an interest in Math. It contains much of the history that lead up to the decision to base significant portions of the soundness of mathematics on top of set theory (and by proxy: Cantor's work on infinities). My recollection is fuzzy since it was years ago but I recall it starts at Zeno's paradox and follows along to calculus and beyond.

The book suggests there was a lot of displeasure and argumentation within the philosophy and math communities because it was felt that there was no real basis for infinitesimals. Some mathematicians (I believe Hilbert and Frege among them?) became determined to shut-up the pesky philosophers by proving the soundness of math based on some logical axiomatic fundamentals. Of course, this was later proven to be impossible by Gödel but at the time they considered it a win that philosophers and mathematicians could at least agree on logic (and more broadly "logical empiricism" which is a basis of "analytical philosophy").

I recall being completely dissatisfied at the arguments presented in favour of ZFC (not mathematically, but philosophically). I remember there was a single paragraph somewhere in the final third of the book that I head to re-read several times before I finally gave up in frustration. My impression of this history is that the mathematicians "won" in some sense by railroading their ideas. Calculus works, right? It is extremely effective and leads to correct results ... so ignore the seeming paradox of summing an infinite quantity of infinitesimally small values and move on already! Further, ignore the actual paradoxes inherent in infinite sets. And this was all done not because there was some problem to be solved but rather to shut-down debate that seemed to undermine the philosophical position of logical empiricism.

Another interesting (if historically questionable) exploration of this topic is the graphic novel Logicomix [2]. This work follows Bertrand Russel and Wittgenstein through this period in our history.

1. https://www.amazon.com/Philosophy-Set-Theory-Introduction-Ma...

2. https://en.wikipedia.org/wiki/Logicomix