Allegedly: the existence of a natural number and successors does not imply, without the Axiom of Infinity, the existence of an infinite set.

Question

The Claim:

From a conversation on Twitter, from someone whom I shall keep anonymous (pronouns he/him though), it was claimed:

[T]he existence of natural numbers and the fact that given a natural number $n$, there is always a successor $(n+1)$, do not imply the existence of an infinite set. You need an extra axiom for that.

It was clarified that he meant the Axiom of Infinity.

The Question:

Is the claim true? Why or why not?

Context:

I like how, if true, it goes against the idea that, if you just keep adding one to something, you'll get something infinite.

This is beyond me. Searching for an answer online lead to some interesting finds, like this.

To add context, then, I'm studying for a PhD in Group Theory. I have no experience with this sort of foundational question. I'm looking for an explanation/refutation.

To get some idea of my experience with playing around with axioms, see:

What is the minimum number of axioms necessary to define a (not necessarily commutative) ring (that doesn't have to have a one)?

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I have included peano-axioms as Peano Arithmetic seems pertinent.

Answer 1

One way of thinking of this is in terms of Peano Arithmetic (PA). The theory PA includes, of course, the successor axiom, and therefore a way of generating arbitrarily large numbers. However, PA does not prove the existence of an infinite set. In fact, PA is equiconsistent with the theory ZFC with the axiom of infinity replaced by its negation; see wiki on this.

Answer 2

As noted in the comments, the structure ($V_\omega$, $\in$) satifies all the axioms of ZFC except for the axiom of existence of an inductive set, and moreover every object of it is finite under all (usual?) definitions (since choice holds), so it follows that some axiom form of infinity is in fact necessary to prove existence of an infinite set

edit to further clarification: while naïvely one may think "but of course there are infinitely many different objects available (inside such a model / provably in such a theory, etc.)", the point is that 'finite' and 'infinite' really are formal phrases; compare and contrast Skolem's paradox

Answer 3

The answer of ac15 is correct and to the point. It helps to clarify the underlying philosophical terminology. Namely, there are two types of infinity, in a sense.

If one says Take $n$, form its successor $n + 1$, and its successor $(n + 1) + 1$ in turn, and proceed, the described non-ending process may be viewed as a potential infinity. This is something that never ends without being a concrete whole. Other examples include when defining a formal theory, you may think of the set of syntactic variables as being potentially infinite (there are always more variables than you need). Or you may believe the tape in Turing machines to be potentially infinite (tape never runs out but does not need to be actually infinite in size), and so on.

When one adds This does not imply the existance of an infinite set, they are talking about completed or actual infinities. The set of natural numbers taken as a whole is (has the property of being) a completed infinity, as are $\mathbb{R}$, $\aleph_{5}$, and so on.

If you lived in $\mathrm{V}_\omega$, you would see $\omega$ as a potential infinity. Instead in $\mathrm{V}$, the set of naturals $\omega$ is the smallest completed infinity. In this particular case, the two notions of infinity are also more-or-less encapsulated by saying that $\omega$ is the trivial strongly inaccessible cardinal, provided you omit the condition of uncountability.