Are there concepts of mathematics today that aren't on a solid logical foundation? In this answer, Arturo wrote:

I would think like Sebastian, that most "working mathematicians" didn't worry too much about Russell's paradox; much like they didn't worry too much about the fact that Calculus was not, originally, on solid logical foundation. Mathematics clearly worked, and the occasional antinomy or paradox was likely not a matter of interest or concern.

And I wondered: is there some mathematical concept today that is not in a solid logical foundation?

Well, we don't know whether $\mathsf{ZFC}$ is consistent, so I guess modern set theory is on pretty shaky ground!
