# Is there a formal distinction between potential and actual infinities?

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?

• It is just this mistaken view, uttered in an upvoted answer there, that I wish to unveil as such. Evidently there is an abyss between pot. and act. and by no means mathematical identity. For instance it is impossible to define a real number by a potentially infinite sequence of digits. – Wilhelm Feb 2 '18 at 16:11
• @Conifold: I think that a farther developed language can define a more primitive one. The language of ZFC is even able to reject the axiom of infinity at all. Further ZFC does not really define actual infinity. At least the axiom of infinity does not yield aleph_0. – Wilhelm Feb 3 '18 at 12:38
• @sand1: In ZFC the axiom of infinity asserts potential infinity. To interpret it as actual infinity is unjustified although it may be argued with the axiom of extensionality. I completely agree with you and Hilbert that for ananlysis potential infinity is fully sufficient. It seems that Zermelo and Fraenkel have been wrong there: "Those who are really serious about rejection of the actual infinite in mathematics should ... do without the whole modern analysis" (Zermelo). "If the attack on the infinite will succeed ... only remnants of mathematics will remain" (Fraenkel). – Wilhelm Feb 3 '18 at 12:56
• @Stella Biderman: There is an infinite set given by the axiom of infinity. The only question is how to distinguish the potential infinity required in analysis from the actual infinity of set theory. That is a matter of interpretation - not the fact that mathematics is heavily based on infinity. – Wilhelm Feb 4 '18 at 9:38
• It took me 10 seconds' worth of reading the Wikipedia entry to be able to summarize this controversy as "whole lot of wanking going on." I categorically deny this argument has any functional value within mathematics. – Carl Witthoft Feb 5 '18 at 13:37

## 4 Answers

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.

• Thank you, but many "modern mathematicians" will not understand this, or disqualify it as purely philosophical. Unfortunately most of them have lost the ability to understand other than formalized text. – Wilhelm Feb 2 '18 at 16:17
• @Wilhelm Can you provide a basis for this statement? – José Carlos Santos Feb 2 '18 at 18:29
• @ José Carlos Santos: With pleasure, more than many. But space is limited. See for instance the upvoted answer in math.stackexchange.com/questions/1351529/…: [They] are mathematically indistinguishable. Or P.L. Clark in hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, p. 11. Or Dan Christensen: There is no need to formally distinguish actual and potential infinity. It never comes up in real-world mathematics. groups.google.com/forum/#!topic/sci.logic/-hsEMkwt6FM%5B1-25%5D – Wilhelm Feb 3 '18 at 12:14
• @Sella Biderman: Mathematicians who understand Aristotle, Cantor, Fraenkel, Zermelo, Hilbert, Nelson, Feferman, Robinson, Jech, Schechter, etc. cannot think that there is no need to distinguish infinities. "In spite of significant difference between the notions of the potential and actual infinite, where the former is a variable finite magnitude, growing above all limits, the latter a constant quantity fixed in itself but beyond all finite magnitudes, it happens deplorably often that the one is confused with the other." (Cantor) ctd. – Wilhelm Feb 4 '18 at 10:02
• For instance, a function could never show countability of a set or its uncountability unless the domain is finished or complete. Simple example: The diagonal argument fails in potential infinity because you never reach the completeness required for a decision. – Wilhelm Feb 4 '18 at 10:05

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.

• Thank you, but I need a formal statement in order to convince modern set theorists. By the way, the actual infinite is not built into the Peano axioms. So what part of Zermelo's axiom "exists { } and with $a$ exists {$a$}" yields finished infinity? Hilbert was enchanted by actual infinity but at the end of the paper praising Cantor's paradise he summarized, somewhat cooling down: "The infinite is nowhere realized; it is neither present in nature nor admissible as the foundation of our rational thinking." I for my person prefer rational thinking. – Wilhelm Feb 3 '18 at 13:10
• @Wilhelm What convinced most mathematicians was reconstruction of mathematics based on set theory with Cantor's actual infinities. Constructivist reconstructions are too restrictive to be widely appealing, although Rodin has an interesting new approach. Finished infinity is given by "there exists inductive set", or something similar. Peano axioms combined with intuitionistic logic presumably express potential infinity only. – Conifold Feb 3 '18 at 23:00
• "There exists an inductive set" alone does not yet make infinity finished. It could also have been said by Peano. But all that would be much easier to discuss if we had a formal definition of what we may colloquially express as: A set $S$ is pot. inf. if "every subset has a proper superset in $S$". A set $S$ is act. inf. if "not every subset has a proper superset in $S$ ". – Wilhelm Feb 4 '18 at 9:48
• @Wilhelm Peano arithmetic does not allow talk about sets, but as long as the logic is classical its countable infinity is presumably actual. Your proposal is called Dedekind infinite, it comes apart from the inductive conception in models of ZF. – Conifold Feb 4 '18 at 22:34

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.

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).

• There is something wrong with the definition: if A is allowed to be C then no set satisfies it, C is not a proper subset of anything, if A is not allowed to be C then every set satisfies it, one can always take B=C. – Conifold Feb 6 '18 at 7:01
• @Conifold: It is tricky. A subset $A$ or $B$ cannot be the potentially infinite class $C$ because the latter does not exist as a completed entity (= set). Only if $C$ is finite or actually infinite, then $A = C$ is possible and the criterion gives the answer not "potentially infinite", i.e., finite or actually infinite. – Wilhelm Feb 6 '18 at 7:52
• It is circular. You are using the distinction which you are trying to define to make the definition. – Conifold Feb 6 '18 at 7:59
• @Conifold: I am not trying to define potential infinity. I am giving a formal criterion. Of course I give it so that the distinction that is usually described colloquially is resulting. – Wilhelm Feb 6 '18 at 8:03
• That is well and good, but so far you defined nothing and gave no formal (or any) criterion. For a criterion one should not have to consult you on what constitutes or does not constitute a "completed" subset before it is "formally" spelled out. At best, you have some vague intuition. You can try to do something inductively, but I see no obvious way to unwind your circle. Or you can try to embed this into some list of axioms implicitly describing relations between finite, potential and actual infinite. But you'll need a lot more axioms to make it viable. – Conifold Feb 6 '18 at 8:22