It is not a mathematical quibble but a philosophical one which means it could end up telling mathematics what to do, after all philosophy still trumps mathematics by being the arbiter of what constitutes true correspondence. So what is actually in play here is what a “set” means, not what infinity means.
In terms of set theory, we make the leap from 1, 2, 3, … to {1, 2, 3, …} via the axiom of infinity, which says that an infinite set exists. It’s much more useful to allow infinite sets so we generally accept the axiom of infinity.
I can now see how Serendipper considers this as strictly speaking an unwarranted shortcut. But I don’t contest that it is useful. Shortcuts are very often useful, look at the Panama Canal.
But if you deny the mathematical existence of the set of natural numbers, yet allow for infinitely many natural numbers, that seems like a philosophical quibble that you would need to justify.
I personally don’t deny the mathematical set. I just deny the philosophical set. I mean I deny that this set of infinitely many numbers has any meaning outside of how the set is being made useful.
And again, note that NONE of this has anything to do with the physical world. Numbers are abstract.
I know that. So the question is how far we want to allow mathematics to operate in defiance of physics.
Because philosophy is about reliability and not about speed. The power gained from seeing sets as potentially having infinite size may come with a drawback of making it doubtful if sets can be trusted, if they can still logically correspond to another set.
I guess what I mean is all questions like, how does the set of integers correspond to the set of real numbers? Does the fact that the second is infinity squared make the former into the root of infinity? If this can’t be addressed there is a logical problem with the infinite set, even if it can still do mathematical work, creating a subprime mathematics bubble.
By what philosophical principle can you say that infinitely many numbers 1, 2, 3, … exist, but that I’m not allowed to put set brackets around the list?
You’re allowed, but all you can do is use that for a specific mathematical operation. That you put brackets around it to mean it goes on infinitely does not mean it actually goes on infinitely. What do I mean by actually, if not physically?
I mean in the sense that it identifies infinity. Thats where philosophy begins to not be bored.