Can we formally distinguish between actual and potential infinities?

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?

• They've pointed out the basic example: a finitist might accept that for every natural number $n$ (whatever that means to them), $\{ 1,2,\dots,n \}$ exists, but for them $\mathbb{N}$ itself doesn't exist. So in this framework, all sets are finite but there is nevertheless no bound on the size of a set. – Ian Jul 6 '15 at 16:40
• "It all seems a bit hand-wavy and circular to me" - it is. – Zev Chonoles Jul 7 '15 at 19:10

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.

• Actually, in the conventional axiomatic system (Peano arithmetic) these sets are not necessarily equal, it has models where the first set is strictly greater because it contains non-standard integers. And the idea that the intended model is standard can not be conventionally axiomatized. It is generally unclear why a mathematical distinction needs to be expressed within axiomatic system, after all we express the difference between standard and non-standard models in meta-language. – Conifold Feb 3 '18 at 7:50

There is a simple distinction: Actually infinite sets are their own subsets. Classes which are infinite but cannot be sets cannot be their own subsets either. They are not complete but they are infinite. That is the definition of potential infinity. We can state If $A \subseteq A$ then $A$ is finite or actually infinite.Otherwise $A$ is potentially infinite.

• What do you mean by "complete"? – Tobias Kildetoft Feb 22 '18 at 10:48
• All elements are existing. – Hippasos Feb 22 '18 at 10:52
• "all elements are existing" is a meaningless statement. – Tobias Kildetoft Feb 22 '18 at 10:53
• No. It is Cantor's finished infinity. – Hippasos Feb 22 '18 at 10:56
• No, seriously, it makes no sense mathematically (not really grammatically). – Tobias Kildetoft Feb 22 '18 at 10:57

from "Infinity and the Mind" by Rudy Rucker, page 89: "Conway [..] even gets a definition of the traditional symbol $\infty$, for potential infinity. $\infty$ is defined as the gap between the finitely large and the infinitely large surreal numbers, and Conway derives the weird equation

$$\infty = \sqrt[\Omega]{ \omega }$$

which magically ties together potential infinity $\infty$, the simplest actual infinity $\omega$, and the Absolute Infinity $\Omega$."