Is there a formal distinction between potential and actual infinities?

Question

In modern set theory the difference between actual infinity and potential infinity is often not understood or even denied. Some decades back however mathematicians like Hilbert or Poincaré, let alone Cantor or Fraenkel were fully aware of the difference. My question: Does there exist a formal definition of potential infinity in contrast to actual infinity, and who was the first to give it?

Answer 1

Not "formal" but quite precise: Aristotle and apeiron.

See Meta, Book IX ($\Theta$), 1048b10:

The infinite and the void and all similar things are said to exist potentially and actually in a different sense from that in which many other things are said so to exist, e.g. that which sees or walks or is seen. [...] But the infinite does not exist potentially in the sense that it will ever actually have separate existence; its separateness is only in knowledge. For the fact that division never ceases to be possible gives the result that this actuality exists potentially, but not that it exists separately.

And Phys, Book III, 4, 206b17:

By addition then, also, there is potentially an infinite, namely, what we have described as being in a sense the same as the infinite in respect of division. For it will always be possible to take something ab extra. Yet the sum of the parts taken will not exceed every determinate magnitude, just as in the direction of division every determinate magnitude is surpassed and there will always be a smaller part.

And 207a32:

It is reasonable that there should not be held to be an infinite in respect of addition such as to surpass every magnitude.

Answer 2

The distinction between potential and actual infinities was Aristotle's clever solution to Zeno's paradoxes. The idea was that while we can mentally divide segments in half indefinitely actualizing the resulting sequence, which is what Zeno does in Dichotomy, is in error. From Metaphysics VIII.8:

"Though there are infinitely many halves in a continuum, these are potential and not actual... So the reply we have to make to the question whether it is possible to traverse infinitely many parts... is that there is a sense in which it is possible, and which it is not. If they exist actually, it is impossible; but if they exist potentially, it is possible."

However, after the invention of calculus alternative resolutions became available, and Cantor's work convinced many mathematicians that other "paradoxes of infinity" could be dealt with as well, see Why did Cantor (and others) use $\mathfrak{c}$ for the continuum?. Moreover, Cantor developed a theory of actualized infinities which was seen as fruitful (Hilbert's "no one shall expel us from the Paradise that Cantor has created" is oft-quoted). The actual infinity is built into the standard axiomatizations of set theory, therefore it can not be distinguished from potential infinity within them. To do the distinguishing one needs alternative axiomatizations of mathematics. This underscores Hilbert's idea of axioms as implicit definitions of terms.

The underlying conceptions were developed by intuitionists, mainly Brouwer and Weyl, see Brouwer and Weyl: The Phenomenology and Mathematics of the Intuitive Continuum by Atten et al. Poincaré, Borel, Baire, Lebesgue and other so-called proto-intuitionists anticipated these ideas earlier, see Did Poincaré say that set theory is a disease? (not exactly). Here is Weyl's informal description of his and Brouwer's conception in Philosophy of Mathematics and Natural Science (1949):

"The notion of sequence changes its meaning: it no longer signifies a sequence determined by some law or other, but rather one that is created step by step by free acts of choice, and thus remains in statu nascendi. This ‘becoming’ selective sequence represents the continuum, or the variable, while the sequence determined ad infinitum by a law represents the individual real number falling into the continuum. The continuum no longer appears, to use Leibniz’s language, as an aggregate of fixed elements but as a medium of free ‘becoming’".

In Das Kontinuum (1918) Weyl undertook an intuitionistic reconstruction of classical analysis based on these infinities in becoming. Weyl's reconstruction was first of many. In 1930 Heyting formalized intuitionistic logic, which allowed to give full formal expression to Brouwer's and Weyl's potential infinities. It turned out that it is not enough to alter set theory axioms (in particular, axioms of infinity and choice have to be dropped), but to also drop the law of excluded middle, which allows reasoning by contradiction. Indeed, if the infinite is never "completed" certain statements about it can not have truth values one way or the other. More recent developments in this direction, like Bishop's constructive analysis, usually go under the name of constructivism (to avoid Kantian associations), see also Constructive set theory. While a minority position among mathematicians constructivism proved to be a lasting presence.

Answer 3

A continuous function can be characterized as a function where the output can be determined to any finite accuracy using only finite information about the input. (By finite information, I mean accuracy. Stupid English language!)

Whether an infinity is "potential" or "actual" depends exactly on what you do with it. A real number is an infinite object; it's got infinitely many digits. Certain operations on real numbers are feasible, whereas others are not. For instance, it's clearly not feasible to decide if a real number is equal to $0$, because that would require knowing every digit of the number. It is feasible to multiply a real number by $2$, because this can be determined to accuracy $\epsilon$ by reading only finitely many digits of the input.

Whether an operation is feasible can be captured either by computability or continuity. Topology and computability theory define the notions of continuous function and computable function. If a function is discontinuous or uncomputable then you've got an actual infinity. Also, continuity and computability are fairly similar in practice.

Answer 4

This is not an answer but too long for a comment. It shows the direction where the answer should be looked for.

A collection or class $C$ is potentially infinite if and only if for every subset $A$ there is a subset $B$ such that $A$ is a proper subset of $B$. Otherwise $C$ is finite or actually infinite.

$C$ is potentially infinite $\Leftrightarrow \forall A\subseteq C$ $\exists B \subseteq C:A\subset B$.

Remark: The complete collection or set of all natural numbers does not exist in potential infinity. The complete real interval $[0, 1]$ does not exist in potential infinity (for instance, because the complete collection of unit fractions does not exist).