I agree. The practical intuitive way to think about “infinity” I use is to simply use a variable N that can be arbitrarily large. And then reason about f(N). It stops you from asking nonsenses like “what is infinity divided by infinity?” since you have to explicitly use N when asking the question. As in “what is f(N)/g(M) for arbitrarily large numbers N and M?”. Which gives you a straightforward way to answer the question.

To be fair, mathematics are not really concerned with the real, physical world. What’s interesting is that we can built axiomatic rules allowing us to properly define infinite sets and these constructions display surprising properties. It’s all purely abstract but it’s a fun thought exercise if you enjoy that kind of things. Plus it has practical applications sometimes which is good I guess.

Don't let the assholes here gaslight you into thinking that Finitism isn't a somewhat hetrodox but still respected field of math. A bunch of great mathematicians historically have felt this way, and a few great ones today still do.

Even if you're willing to accept the idea of cardinalities of infinity (and I think you ought to), I find that the more broad acceptance of 1. The axiom of choice (vs the axiom of determinancy, it's opposite) and 2. The law of excluded middle to be highly suspicious.

If you reject the axiom of choice, you're just in alternative but still correct math.

If you reject the law of excluded middle, you've ended up with intuitionistic or "constructive" logic, where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality. That is, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied, but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality.

Notably, in intuitonistic/constructive logic, infinity is rejected until it can be "constructed", which also means that Cantors diagonalisation argument is not so naively accepted. While that diagonalisation itself was constructive, Other related and stronger theories from Cantor are not.

Well if we're going to start deciding what things to believe in based on whether they're comfortable or not I am going to stop believing in traffic and the texture of coconuts.

For me, I stopped thinking of "infinity" meaning "bigger number than you can imagine" to just simply meaning "unlimited" or "never ending."

That way the term works equally well at different scales. There are infinitely many real numbers between 1 and 2. There are infinitely many natural numbers.

That's quite uncontroversial. But this is about math.

How do you reconcile the fact that the best tool we have, by far, to describe the "real, physical world" depends on certain notions of infinity?

Do you believe in 0?