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Sure, that way lies intuitionism, which is a perfectly respectable (if not particularly popular) school of mathematics. > Once you think you can reduce somethi

Sure, that way lies intuitionism, which is a perfectly respectable (if not particularly popular) school of mathematics. > Once you think you can reduce something unknowable, to a symbol, you can prove anything. That doesn't follow. If ZFC is trivially inconsistent then someone would probably have noticed by now and proved P && !P for some P and brought it crashing down. That hasn't happened though, so even if you have strong aesthetic preferences against infinite sets being actually real, it seems like you can treat them as if they are real and produce a productive and not-obviously-inconsistent mathematical system. Most mathematicians don't really care if the naturals are actually infinite or just can be productively treated as if they are.

nico 4y ago
> That doesn't follow. If ZFC is trivially inconsistent then someone would probably have noticed by now and proved P && !P for some P and brought it crashing down. Ahh, the classic economic argument, “that’s not a $100 bill on the sidewalk, because if it was, someone else would have picked it up already”. That’s a great way of accepting everything blindly to justify not questioning things. It also means you are using popularity as a measure of truth. It’s fine if mathematicians don’t care if naturals are “actually infinite”, but then what’s the point in reasoning about infinity through math if technically you are not really saying anything, but you’re going in circles around your own definitions instead?