Poincaré was a genius but math has moved forward since the 19th century.

Here's what I'd argue is the most important take away from set theory: for a given framework of mathematics, there are conjectures one can make that can be neither proved nor disproved. Perhaps the most high-profile of them is the continuum hypothesis.

Maybe P!=NP is such a conjecture. Maybe the existence of a solution to the Navier-Stokes equation is such a problem. For Poincaré, the very idea that there is math that can be neither proved nor disproved was not in his mental vocabulary.

On another note, Weil's celebrated proof of Fermat's last theorem relies on the existence of inaccessible cardinals[1], although I've heard it conjectured that they are not necessary.

[1] https://en.wikipedia.org/wiki/Inaccessible_cardinal

Well, I've studied under a pupil of Grothendieck, so, yes inaccessible cardinals are at the heart of his category theory. When you study this long enough, you will discover they essentially perform the function of what in computer science is the class concept (which why these structures were called 'universes'). But Russell already developed a type theory around 1900. Most mathematicians have never cared to really study Gödel for instance. Because you then would have to read Russell's principia mathematica and Frege's concept script.

A minor nitpick: It was Andrew Wiles who proved Fermat's last theorem. Andre Weil was another incredible mathematician (responsible for, among many other things, the celebrated Weil conjectures), but did not prove Fermat's last theorem.

Andre Weil was very opinionated and lived a very interesting life. I encourage you to read about it: http://en.wikipedia.org/wiki/Andr%C3%A9_Weil