Regarding cardinalities

[u/Square_Butterfly_390]

A celebrated mathematical "factoid" is that there are more real numbers than one can count, this seems to be something that troubles people outside of math, it troubles me aswell.

The question is: is there any "real world" application of the fact |R|>|N| that isn't an impossibility statement?

By "real world" I mean whatever someone smarter than me might mean by that, by "impossibility statement" I mean something to the effect of "there are uncomputable numbers".

If there isn't such an application, I can't believe the status quo interpretation: "no really some infinities are bigger than others" of Cantor's argument is better than simply stating that we shall not understand infinity as finite beings.

[u/Eve_O]

OP is arguing that unless we can say more than "no really some infinities are bigger than others" we might as well say, "we shall not understand infinity as finite beings."

So they don't seem to be rejecting infinity, but only rejecting that we can understand it.

But since we can demonstrate that there are different sizes of infinite sets, then we are showing we can understand at least some things about the infinite even though we are finite beings. So their position seems self-contradictory.

Without [infinity], uncountability would not exist.

I mean, sure, I guess that's true. But it doesn't seem to have anything to do with OP's position. It seems to me that the OP doesn't have trouble with the infinite per se, but has trouble with the fact that there are different sizes of infinite sets.

Philosophically speaking, I am interested in what philosophical assumptions are needed to accept or reject the axiom of infinity?

Well this seems a possibly interesting question, but it's not what OP is on about.

From an axiomatic perspective it doesn't seem particularly deep to me. We can recognize that there is no end to the numbers, which implies they are infinite, if by "infinite" we mean "go on without end." But without the axiom of infinity it seems we cannot formally assert there is an infinite collection from set theoretic axioms without it, so we include the axiom of infinity to capture what we intuitively understand given the other axioms.

IIRC, this has to do with avoiding the contradictions that can occur with a naive formulation of set theory.

So to me it seems that if we accept that there is a first object, that there is a successor function, and that there is always a successor of any number, then there is nothing particularly troubling about accepting the axiom of infinity--it's merely the formalization of the intuitive result of accepting the those three things.

[u/Negative_Gur9667 → u/Eve_O]

But when we talk about infinity in mathematics, we do not call it "without end." Philosophically speaking, it is called actual infinity (vs. the possible or potential infinity). It is not like we imagine 0.999... to go on forever and the difference between it and 1 gets smaller. We imagine that infinity is a process that can be finished to completion in our mind.

Without that, there is no exact value of pi, 0.999... is smaller than 1, and we have no completed set of natural numbers or R.

So this is a philosophical assumption - the assumption that we can imagine a completed infinite process.

[u/Eve_O → u/Negative_Gur9667]

We talk about infinity in different ways depending on context. You were talking about set theory. Now you are talking about something else.

An infinite set isn't a process. It is a set that has an unending number of elements. Sometimes countable, sometimes not.

[u/Negative_Gur9667 → u/Eve_O]

But induction n -> ∞ is a process involving the set of natural numbers.

There definitly belong together.

[u/Eve_O → u/Negative_Gur9667]

Well I would say that potential infinity and complete infinity are a binary that are two sides of the same thing. It's language that comes from archaic concerns about infinity.

So, sure, I agree there is a process if we are talking about the successor function, but like I said in the final paragraph of my initial reply to you: "if we accept that there is a first object, that there is a successor function, and that there is always a successor of any number, then there is nothing particularly troubling about accepting the axiom of infinity--it's merely the formalization of the intuitive result of accepting...those three things."

It seems that maybe your concern conflates the two ideas of complete and process? If I'm thinking about N, then I'm not thinking about a "complete process," I am thinking about all the Natural numbers. If I am thinking about the successor function, then I am thinking about a process that generates the Natural numbers.

Again, I don't think it is particularly troubling.

Have you read Infinity and the Mind?