# Is actual infinity physical infinity? Or just the axiom of infinity?

I've always been a little confused on this point.

My (second-hand) understanding of Aristotle's difference between potential and actual infinity is this:

We all have an intuition of the counting numbers 1, 2, 3, 4, ... The idea that "there's always a next one" is encapsulated by the principle of induction, which says that "if n exists, so does n + 1".

That's potential infinity. There are infinitely many things, in the sense that there's no end to the list. We can go as high as we want. But there is no completed "set" of all those numbers.

On the other hand, the axiom of infinity says that there is a set, which is a thing that satisfies the axioms of set theory, that contains 1, and whenever it contains n it contains n + 1. [I don't care if you prefer to start counting with 0, makes no difference to this point].

So to me, potential infinity is induction; and actual infinity is the axiom of infinity. An ultrafinitist rejects induction; a finitist accepts induction but rejects the axiom of infinity; and an infinitarist (not a standard term) accepts both induction and the axiom of infinity.

Now I have also seen "actual infinity" meaning physical infinity: the idea that there might be infinitely many planets, stars, electrons, intervals of time, "causes," etc. One sees this usage in William Lane Craig's theology, pointing out that an "actual infinity," by which he means a physical infinity, must be absurd because it would be subject to the "paradox" (which is not really a paradox) that an infinite set can be placed in bijection to one of its proper subsets, as in Galileo's paradox or Hilbert's hotel.

I am wondering what Aristotle had in mind about actual infinity. Whether he meant physical infinity, or just a conceptually completed collection containing all the natural numbers.

And secondly, is there a standard set of definitions in philosophy to disambiguate these terms, such as "actual infinity" versus "physical infinity," where the former means abstract sets whose existence depends on the axiom of infinity, and the latter means an infinite amount of physical stuff.

Thanks for any clarity on this issue.

• See Aristotle on infinity for some details : "Aristotle argues that in the case of magnitudes, an infinitely large magnitude and an infinitely small magnitude cannot exist. In fact, he thinks that universe is finite in size. [...] However, since Aristotle believes that the universe has no beginning and is eternal, it follows that in the past there have been an infinite number of days. Hence, his rejection of the actual infinite in the case of magnitude does not seem to extend to the concept of time." – Mauro ALLEGRANZA Jun 17 '18 at 19:20
• This paper by Shapiro and Linnebo goes over all aspects of your question very well. I haven't been able to find a preprint version of it, but here is Shapiro giving a talk about the paper and here are the (extensive) slides for the talk. – Not_Here Jun 19 '18 at 7:21
• Hamkins and Linnebo also write up a very interesting technical paper expanding the topic. – Not_Here Jun 19 '18 at 7:29
• @PeterJ Thanks much. You seem to be the only one who understood my question. "Much confusion is tolerated." That sums it up. Thanks again. – user4894 Jul 30 '18 at 16:57

Actual infinity has been created by Cantor in order to describe physical infinity.

"I refer you to what I have found in Math. Annalen Vol. XX pp. 118-121, that in the space filled with body matter (since I assume the body matter being of first cardinality) for the ether (the matter of second cardinality) there is an enormous space remaining for continuous movement, such that all phenomena of transparency of bodies as well as those of radiating heat, the electric and magnetic induction and distribution appear to get a natural basis free of contradictions." [G. Cantor, letter to G. Mittag-Leffler (16 Nov 1884)]

Not necessary to mention that modern science has no use for ether and neither actual infinity nor the theory based upon it, namely transfinite set theory.

• I agree that Cantor may have had physical space in mind. But my understanding is that today we distinguish mathematically infinite sets from an actual physical infinity. We can have all the alephs in our math even if the universe turns out to be finite. Just as we can invent the game of chess even though there are no physical referents for the concepts of chess. It's just an abstract game, like infinitary math. So actual infinity has a dual meaning: the completed infinities given to us by the axiom of infinity, which are abstract; and the idea of an actual infinity in the world. – user4894 Sep 15 '18 at 2:58
• ps -- When you wrote that modern science has no use for ether; did you mean that literally? Or was that a typo operating as a play on words, intended to be "either" but also a pun on the luminiferous aether? – user4894 Sep 15 '18 at 2:59
• There are two differences between actual infinity and chess. First the rules of actual infinity are self-contradictory. To see this consider the natural numbers as the simplest example: Every natural number that I can refer to belongs to a finite initial segment that is followed by potentially infinitely many natural numbers. An infinite set is much larger than every finite set. Therefore almost all natural numbers cannot be referred to individually. Nevertheless set theory claims that I can refer to every natural number. Contradiction. If you are not convinced, try it! Second ... – Wilhelm Sep 15 '18 at 11:03
• All positions of chess can be represented on a chess board. An infinite set, let alone an uncountable set, cannot be represented because the universe has less than 10^100 bits available. And even neglecting physical constraints: There are only countably many finite strings available - therefore most real numbers are undefinable. Modern set theorists have swallowed the concept of undefinable "real" numbers. Cantor, who was not yet immune to those ideas, opposed vigorously (Letter to Hilbert, translated in hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf, p. 27). – Wilhelm Sep 15 '18 at 11:10
• Mathematical infinity and chess are similar in that they are formal systems that do not necessarily refer to anything in the real world. That's the only analogy I made. I don't understand why you wrote two lengthy comments to argue that they're different because chess is finite. That's irrelevant to the point I made. And you didn't engage with my main point, which is that what Cantor believed historically is not how we view things today. – user4894 Sep 16 '18 at 0:27

I think you might be a little confused from your second hand reading of Aristotle; he doesn't describe potential infinity as formal mathematical induction; what he does say is that potential infinity is best described as saying however much you hold or describe it, it is always larger.

Take for instance, your first 'completed' infinity: 1, 2, 3 ... omega; but can it really be described as a completed infinity when directly after this you have omega+1, omega+2, omega+3...?

As you can see from this, your so called completed infinity is not a complete infinity because the series still carries on.

• Perhaps I wrote too much in my question. I don't really care about Aristotle except that he originated the potential/actual distinction. I'm more curious to know if there is standard terminology to disambiguate an actual infinity occurring in the physical world with the actual infinity of the natural numbers. The transfinite ordinals don't apply at all here. The axiom of of infinity gives you a "completed" infinite set. I'm not wrong about that. A finitist for example admits induction, but denies the axiom of infinity. The transfinite cardinals are a red herring here. – user4894 Jun 19 '18 at 17:58
• @user4894 if you're interested in the idea of the axiom of infinity as provided a "completed" set then "completed" and Aristotle's "actual" are different. What the axiom of infinity states is that there exists a set that is potentially infinite - that is, that it always contains something more. Even, for example, "omega+1" is another potential infinite - it's more than the thing that always contains more. It sounds like you're not really interested in the idea of a set that is "actually infinite". – Paul Ross Sep 2 '18 at 7:20

Dominic Soto said this about the difference between categorical/actual infinite and syncategorematic/potential infinite:

Modern philosophers (Neoterici philosophi) declare that in respect to continuous magnitudes, the term infinite can be understood in two ways; firstly, it can be taken categorically...; secondly, it can be taken syncategorematically; the meaning of this adverb can be explained by these words: an amount that is never so great that it cannot become more (non tantum quin majus)… In addition, they pose this rule: When the word “infinite” is placed on the side of the predicate of a proposition, it is taken in the literal (nominaliter) and categoric sense, as in these sentences: Deus est infinitus, continuum habet partes infinitas. When, however, the word “infinite” is put side on the side of the subject, it is taken in the syncategorematic and explanatory sense (exponibiliter), as in this proposition: Infinita parva est pars continui.

—quoted in pt. 3 "Dominic Soto & Parisian Scholasticism", § "Potential infinity & actual infinity", of Galileo's Precursors

I believe your confusion comes from not noticing the difference between an infinite number of "physical" things, and an infinite number of numbers (non-physical things)!

If we decide to use the number of electrons in the universe as a way to get a very large number of physical things, although we get a very large number (10^90), it is finite!

If we decide to use the number of numbers (non-physical things) between 1 and 2, there is an infinite number of them!

• +1: Which of course is why Aristotle doubted the possibility of infinite divisibility; that we have such acting as quantum mechanics has borne that out generally. – Mozibur Ullah Jun 19 '18 at 5:33
• I don't believe I'm confused at all. I'm asking if philosophers have standard terminology to distinguish a physical infinity from an abstract one such as numbers. Can you please re-read my question and clarify whether I either misspoke myself or you misread it? – user4894 Jun 19 '18 at 16:02
• ps -- I see that if you only read my title and not my post you would get the impression you did. My question is: Is there a standard terminology to distinguish between an actual infinity of stars and the actual infinity of the counting numbers? There seems to be considerable confusion in the literature regarding this terminology. – user4894 Jun 19 '18 at 16:46