> 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 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.
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?
It’s impossible to have a known cardinality of an “infinite set”.
The definition of set is “a collection of items”. You cannot “have” an infinite collection of items.
You can have a method/function/formula/algorithm that generates as many items as you can in a certain amount of time, but by definition of infinity, you can never “count” an infinite number of elements.
Hence, you can never know the cardinality of an infinite set (like the natural numbers).
Once you think you can reduce something unknowable, to a symbol, you can prove anything.
Like when proving 1=2 by dividing by 0.
But then what’s the point?