Is the set $I$ in the axiom of infinity often interpreted to be an actual infinite set?

Question

After some discussion on philosophy stackexchange I wanted to know what some set theorists consider the infinite set that the axiom of infinity posits to exists.

Question

Do most, or you if you're a set theorist reading this question, set theorists believe that the infinite set $I$ in the axiom of infinity believe that $I$ is a completed infinite set or just a potentially infinite set?

This question is important to me, because I've been wondering how most mathematicians can believe that the set $\mathbb{N}$ is a completed infinite set without the presupposition that actual infinite sets exist. Then I began reflecting on the axiom of infinity, which would provide the mathematician a starting point to begin working on deductions concerning actual infinite sets if this axiom concerns actual infinite sets.

Another reason I'm asking is because I know that if a mathematician is a constructivist or intuitionist, then the axiom of infinity is used as an axiom in their set theory, and I believe this is possible if they interpret that $I$ is a potentially infinite set.

Lastly, and I don't know if it's possible to answer this question, but when Zermelo first proposed the axiom of infinity, did he have in mind that $I$ is an actual infinite set or did he withhold any claims as to whether or not it was an actual or potential infinity?

I'm sorry in advance if this question is not suitable for this stackexchange, but my prior discussion on philosophy stackexchange has led me to confusion as to whether or not $I$ can be interpreted as an actual or potential infinite set, so I was curious to know what some or most mathematicians, especially ones who have no problem with actual infinite sets, consider $I$ to be with respect to the infinite.

So what is $I$ to you?

Answer 1

I, personally, don't even believe that "potential" is a meaningful adjective to attach to "infinity".

Or, more accurately, that the term is a misnomer — when used, what people seem to have in mind is a class of finite things, and there is no upper bound on the sizes of the members of the class.

For example, if someone disbelieved in the infinite divisibility of a Euclidean line and spoke of its point-set as merely being "potentially infinite" set, they aren't talking about its point-set at all — instead, what they are talking about is all of the different you can mark finitely many points on the line.

With the misleading term "potential infinity" discarded, and replaced with the more precisely defined notion of an "unbounded family" of things, the answer to your query is now obvious: the axiom of infinity specifies the existence of one set with a particular property, not a whole family of sets with fragments of that property.

Answer 2

Looking in Gregory H. Moore's book "Zermelo's Axiom of Choice", one can find a brief history of the axiom of infinity (pp. 154--155).

Its roots lie with Bolzano, who argued the existence of an infinite set (via a somewhat-diagonal argument, like the Euclidean proof of the infinitude of primes); Dedekind later reformulated this into what we know now as "Dedekind-infiniteness"; and several others have formulated this in one way or another. However, the foundations of mathematics were still shaping, and a lot of this won't pass the test of modern rigor. Zermelo, however, was different, and postulated the axiom by formalizing exactly what it means for a set of "repeated singletons" to exist.

It seems probable that Zermelo had the natural numbers in mind, since at this point Peano's postulates already existed, and so modeling the natural numbers as $0=\varnothing$ and $n+1=\{n\}$ seemed like a natural thing to do. And Zermelo's axiom of infinity essentially postulated the existence of the natural numbers as a set using this modeling.

The modern formulation of Infinity, however, comes in later, when we think about ordinals, and $\omega$ or $V_\omega$ in the von Neumann hierarchy. To a set theorist, "the infinite set" is by all means $\omega$, which is also the natural numbers for all useful purposes in the context of set theory.

As far as the terms "potential" and "completed" or "actual", I think that you're mixing several philosophical approaches here. In the context of real analysis, sets are not of interest, and only real numbers and functions (or sequences) are of interest. Those are, naively, given by various constructions, and you can ask whether or not something is unbounded. In other words, does a function attain arbitrarily high values? If so, it can be seen as potentially infinite. However every valid input has to return a real number, so it cannot be infinite. Therefore "potential infinity" works just fine.

In set theory, however, we take a fixed universe, and argue over it (using what basic assumptions we made on this universe, e.g. a model of $\sf ZF$ or $V=L$ or whatever). So when we look at an infinite set whose existence is guaranteed by the axiom of infinity, we do not look at it as something "to be constructed in parts", but rather something which already exists. We assumed the axiom, and derived (using existential instantiation) the existence of said set. We can prove that this set is also infinite in any "reasonable" definition of infinite, as far as the universe we work inside cares to admit. Which is why we all have the tacit agreement "yes, this set is infinite".