Can we formally distinguish between actual and potential infinities?

Question

Actual infinity is the idea that numbers, or some other type of mathematical object, can form an actual, completed totality; namely, a set. Hence, in the philosophy of mathematics, the abstraction of actual infinity involves the acceptance of infinite entities, such as the set of all natural numbers or an infinite sequence of rational numbers, as given objects. This is contrasted with potential infinity, in which a non-terminating process (such as "add 1 to the previous number") produces an unending "infinite" sequence of results, but each individual result is finite and is achieved in a finite number of steps.

Source: "Actual Infinity," Wikipedia

From another source:

Potential infinity refers to a procedure that gets closer and closer to, but never quite reaches, an infinite end. For instance, the sequence of numbers 1, 2, 3, 4, ...

Completed infinity, or actual infinity, is an infinity that one actually reaches; the process is already done. For instance, let's put braces around that sequence mentioned earlier: { 1, 2, 3, 4, ... }

Source: Eric Schechter, "Potential versus Completed Infinity",2009

EDIT: How can we talk about an infinite sequence of natural numbers without having already established the existence of the set of all natural numbers using Peano's Axioms or some equivalent?

It all seems a bit hand-wavy and circular to me. Can we formally distinguish both kinds of infinities?

Answer 1

The distinction here is really philosophical rather than mathematical, but I wouldn't say it's hand-wavy at all. In any case, here's my best attempt at an answer.

Suppose we want to make a mathematical distinction between an actual infinitude and a potential infinitude, where by 'mathematical distinction' I mean a negation of an equation between two mathematical objects which is provable from a conventionally accepted system of axioms (e.g. ZFC).

The closest I can think to putting your question into words mathematically is: define sets $\mathbb{N}_a$ and $\mathbb{N}_p$ by

• $\mathbb{N}_a$ is the card-carrying, honest-to-god, set of everything that self-identifies as a natural number;
• $\mathbb{N}_p$ is the set of everything obtainable from $0$ by iterating the successor operator finitely many times.

...but in any conventional axiomatic system, these sets are equal: mathematics only pays attention to extensional equality between sets, even if they aren't intensionally equal. That is, sets may have different descriptions, but if they have the same elements then they're declared equal. As such, these two sets are mathematically indistinguishable.