Is infinity and recursion linked?

Question

noob question, from a programmer with a light background in mathematics.

I was thinking about how one would definite infinity, and came across a thought: how could one define infinity?

One thought was: "the number one bigger than the biggest number so far", moving up and up.

This made me think: how else could one define infinity?

there's 1/x as x approaches 0, but I realize, x approaches 0, may also intuit some kind of recursion: subtract dx from x, over and over again.

So then, the question:

Can infinity be only defined by some sort of recursive process? Is there any other way?

I may be misusing terms above, but it got me curious, about the link between infinity and recursion.

Would love to thoughts / resources on exploring this link

Answer 1

Infinity is a concept, and it can be defined like any concept by a proposition.

For example $\infty := \exists \infty:\forall y: \infty > y$

And you can do the same in the programming language. In fact, you can even continue to transfinite numbers (sort of) by defining for example pair $(n,1)$ which is greater than infinity and hence greater than any number... and continue similarly $(n,2), (n,3)$ etc. Then you could use "transfinite induction" to prove things about this construct. So it can be quite constructive, even...

Answer 2

One way to characterize infinite sets (assuming the Axiom of Choice) is that a set $S$ is infinite if and only if there is a $1$-$1$ map $f: S \to T$ where $T$ is a proper subset of $S$; i.e., $T \subset S$ and $T \neq S$.