I mean, me neither. But if all sets are finite AoC just trivially holds, right? You can do it "manually".

If you back off to just ZF, parts of functional analysis will break. L^infinity^ space isn't separable, and so isn't necessarily Baire anymore, for example.

If we go all the way to finitism or ultrafinitism it doesn't really exist as a concept in the first place. But, whatever numerical engineering calculation will still work, and you can probably do something that looks like functional analysis to determine a mode of vibration, even if you're actually just using a series of high-dimensional but finite spaces. Probably, anyway. Don't ask me to prove it.