From my “watched a YouTube video” understanding of Gödel’s Incompleteness Theorem, a consistent mathematical system cannot prove its own consistency, and any seemingly consistent system could always have a fatal contradiction that invalidates the whole system, and the only way to know would be to find the contradiction.
So if at some point our current system of math gets proven inconsistent, what happens next? Can we tweak just the inconsistent part and have everything else still be valid or would we be forced to rebuild all of math from basic logic?
ah I gotcha, thats fair then.
On another note, you’d propose a von Neumann hierarchy without AoC? How would that class be defined?
Admittedly I’m not that big into class theory so I am likely missing something obvious, but to me it seems like the definition of the class requires such a selecting function
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. Linfinity 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.