Why is there no infinite natural number, and why does finiteness need to be closed under the successor function?
I think can understand why something like $…S(S(0))…$ is not a natural number because it is not the successor of anything. Likewise, I can see why $S(S(…))$ is not a natural number because it is its own successor.
However I might be able to imagine some strange sequence if adding successor functions to the outside and inside such that you can’t make these same arguments.
One of the big issues that I see is that the “…” parts which represents an infinite sequence of S() only seem to make sense if you define what infinite actually means, and thats typically defined as something that is not in bijection with a set $\{0,1,…,k\}$ for any natural number, $k$. While this is obviously means that our proposed number is not a natural number, it seems to just be circular since what we really want to encapsulate is the “foreverness” of this object.
One way to do this is to say that $0$ is finite and that if $a$ is finite then so is $S(a)$.
This resolves the circularness, but I don’t exactly see why the second half of the statement must be true. Does it really break the idea of “foreverness” to have some threshold where everything before and including it is finite and everything after is not?
I suppose what I am asking is this: We have a concept of a smallest infinite quantity. This is usually represented as $ω=\mathbb{N}$.
Is there any meaningful way to define both a largest finite quantity and a smallest infinite quantity? I imagine this would violate ZFC, but the Peano axioms can be formulated without ZFC so its not necessarily an issue.
I imagine this is equivalent to formulating a kind of ordinals with both a successor and precessor function, and then asking if you can create infinitely large ones from a given starting point.
We can use the definition of Dedekind infinite to avoid appealing to "foreverness" or the successor function. To do this we say a set $S$ is infinite if there exists some $S' \subsetneq S $ and a bijection $f:S' \rightarrow S$. A set is finite if it's not infinite. This does characterize the finite sets since if I remove an element from a finite set $S$ then $f$ cannot be surjective, and therefore is not bijective. We can see intuitively that if something is infinite, I can take some away and have the same amount left. That's the idea being used here.