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I could use a little exposition on the significance of the distinction. I'm aware that potential infinities have arbitrarily large numbers, whilst actual infinities refer to the number "infinity" itself. However, I'm a little lost as to why potential infinities seem to be more epistemically esteemed; is it even feasible to have a potential infinity without necessitating an actual infinity? What is the philosophical significance of distinguishing the two?

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This is an extremely broad topic, but there are a few concepts we can introduce to get you started. The terms "actual infinity" and "potential infinity" aren't really used in mathematics, but they do appear to relate to a distinction that is made: the ordinals vs the cardinals. There are also a variety of sizes of infinity within each category, that we will briefly discuss.

A note on typography: Unfortunately, philosophy.stackexchange doesn't seem to support typesetting mathematics (if there's a way to do it, someone let me know). So assume A_i means A with the subscript i. A_{i+1} means A with the subscript i+1.

Types of Infinities

Let's examine each of these using the natural numbers, ℕ, and the successor function from Peano Arithmetic: S(n) = n + 1 for all natural numbers n.

∞ can be used to represent any type or size of infinity, and may show up when the context makes it clear which we're dealing with. But there also exist more precise notations, which we'll be using exclusively here.

Potential infinity

Consider a sequence that has no greatest element, but for which every element is finite. e.g. let A be a sequence such that A_0 = 0 and A_{i+1} = S(A_i). Every A_i is a finite natural number, but there is no greatest element. The elements of A are thus potentially infinite.

Actual infinity

The set of natural numbers, ℕ. There are actually an infinite number of them, not just an arbitrarily large number. This relates to the potential infinity in that ℕ can be defined as follows:

  • 0 is in ℕ
  • If n is in N, then S(n) is in ℕ.

Which is to say, A contains exactly the same elements as ℕ. The only difference between the sequence and the set is that the sequence is ordered. So it would seem that there isn't an especially meaningful distinction between "potential" and "actual" infinities. But there is a meaningful distinction between the size of A itself and the size of the individual elements of A. And that brings us to the cardinals and ordinals.

Cardinals

The cardinals are used for counting the number of elements in a set.
The standard notation for the size of set A is |A|. The value of |A| is always a cardinal. Cardinals can be finite or infinite. e.g. |{1,2,3}| = 3.

If the set is actually infinite, we use the aleph (ℵ) numbers to represent them ℵ_0 is the cardinality of the natural numbers: ℵ_0 = |ℕ|, and is the smallest of the infinite cardinalities.

Ordinals

The ordinals are used to establish an ordering over a set. The ordinals themselves are fully ordered, and thus if we can establish a one-to-one mapping between any set and the ordinals, we can establish a well-ordering over the set. In the example of A, that involves mapping the subscript to the value of the element. Since A_i = i, that's trivial to do: f(i) = i.

For the finite ordinals, we just use the natural numbers, same as for the cardinals, since the context makes it clear whether we're talking about ordinals or cardinals.

There are also infinite ordinals, but they work a bit differently from the infinite cardinals, in that they're defined (at least partially) as limits. The first infinite ordinal, ω, is defined as the smallest ordinal that is greater than all the natural numbers.

ω is the limit ordinal for the set A: A_i < ω for all i.

Summary

So we have our set, A, which we can just call ℕ at this point since that's what it is:

  • ℕ_0 = 0, ℕ_{i+1} = S(ℕ_i).
  • For all i, ℕ_i < ω.
  • |ℕ| = ℵ_0

The idea of "potential" vs "actual" infinities is somewhat hand-wavy, but the basic idea can be put on a more rigorous foundation by differentiating the order of the elements of a set, vs the number of elements in the set.


Sizes of infinities

This section is less relevant to your initial question, so I've split it off from the main answer, but it is important if you want to get a deep understanding of infinities.

Cardinals

Two infinite sets are considered to have the same cardinality if we can establish a one-to-one mapping between them.

For example, the set of all integers, ℤ, has the same cardinality as the set of ℕ, which we can prove as follows: Let a be an arbitrary member of ℤ. If a is negative, we map it to -2a-1. If a is non-negative, we map it to 2a. This results in each element of ℕ mapping to a unique element of ℤ, and vice versa, so we can say the sets are the same size.

On the other hand, we cannot do this with the real numbers. The proof, using Cantor Diagonalization, is a bit long to explain here, but understanding it will go a long way towards helping you understand infinities. The cardinality of the reals may be the next element in the set of cardinals, ℵ_1, but this hasn't been proven. There are an infinite number of cardinalities.

In practice (at least in my field), we can just classify the infinite cardinals as "countable infinities", those of cardinality ℵ_0, or "uncountable infinities", which is every other size.

Ordinals

There are also an infinite number of ordinals. After ω, we have ω+1, ω+2, etc. After all of those, we have ω+ω = 2ω. And so on.

  • this is a helpful addition, thanks – another_name Jan 2 at 7:10
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According to The Logic Museum's page on Philosophy of the Infinite1, the distinction traces back to Aristotle to whom is attributed "infinitum actu non datur" - that actual infinity does not exist2. The details of his views and arguments are complicated enough to be their own questions. However, as a result the idea of potential infinity - something that can continue without implying any end, is taken to be well established in philosophical tradition. In contrast, it's not always clear what it means for something to both be actually infinite and reducible to an aggregate at the same time. In one case, Aristotle seems to argue that we can't expect a line to be actually composed of an infinite number of points (as opposed to potentially dividable into infinitely many segments) when we can't find any two points to be adjacent (and thus able to span the continuum).

In ancient times, this was relevant to things like the atomic theory of Leucippus and Democritus, and Zeno's paradox. But the distinction has been important to many philosophers in a variety of contexts since then (cf. Logic Museum). In part, the esteem of potential infinity over actual infinity is testimony to Aristotle's legacy in western philosophy. For instance, I think many theistic philosophers would argue for both.

In relatively modern times the question saw something of a revival during the development of axiomatic set theory. Dedekind, Cantor, and others addressed the problem around that time, and many questions related to infinite sets (for instance, the continuum hypothesis and axiom of choice) were developed. These became of particular importance since transfinite set theory based on ZFC (Zermelo-Fraenkel axioms plus the axiom of choice) was becoming a viable system in which to formalize modern foundations of mathematics.

Nowadays, transfinite set theory is the most accepted foundation of mathematics. Because of this the idea of actually infinite sets is so common and useful that the question has lost some of its significance. In this context, the idea that a potential infinity implies an actual infinity is plausible at least in a logical or immanent sense. I think the question is still interesting from the perspective of ontology, alternative foundations of mathematics, alternatives to ZFC in set theory, and in other historical and related contexts. There are also some interesting mathematical aspects, for instance in set theory and analysis, but I think that would be better asked on MathOverflow.


1: The Logic Museum's Philosophy of the Infinite has a selection of quotes and some writing on treatments of the infinite by Aristotle and later philosophers.
2: See, for example, Part V in Physics, Book III

  • thank you greg, this is a fantastic answer – another_name Jan 2 at 7:00
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I will answer your question why potential infinities seem to be more epistemically esteemed. The reason is that potential infinity is possible and actual or finished infinity (as it has been called by its inventor Georg Cantor) is impossible.

Potential infinity: 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.

Actual infinity: All natural numbers can be referred to individually.

The latter is obviously impossible if the majority of numbers is always beyond the natural number referred to. You can also verify it experimentally. Refer to any natural number. It belongs to a tiny initial segment as is proved by the fact that multiplying it by an arbitrarily large factor results in a natural number too.

Therefore most philosophers and mathematicians (except the small sect of set theorists) esteem and adhere to potential infinity.

  • i up-voted this answer, because it comes from a position of authority and it's very understandable. however, some references would be preferable -- even as it's covered in other answers – another_name Jan 2 at 7:06
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    @another_name: Are you aware that this is one of numerous sockpuppet accounts of a complete crank called Wolfgang Muckenheim? – user21820 Jan 4 at 17:40
  • oh right -- the profile says he's a professional mathematician @user21820 – another_name Jan 4 at 17:42
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    @another_name: You too can put a statement on your profile saying that you are are professional mathematician. I hope you don't just believe what people claim. Moreover, if you have just a little bit of mathematical training, you can easily see through all the nonsense this crank posts. In particular, this is a crank that denies the correctness of Cantor's theorem that there are uncountably many reals (or equivalently subsets of the naturals). – user21820 Jan 4 at 17:47
  • @user21820 ha i won't do that... just explaining my error, sorry, i have not studied maths. his account should be deleted then – another_name Jan 4 at 17:48
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This requires exacting conceptual clarification even to reach the surface of the conceptions in play, and this prior to seeing what motivations diverse thinkers had in holding their grip on this or that conception. The answer depends on the seeker after the concept and how they seek the concept of infinity. If the essence of infinity is thought as boundlessness, meaning something added to it will never exhaust it, then it is actual in that it is now, right now, able to receive anything to itself without being exhausted. It's clear that this conception of infinity is much different than that which would think the essence of infinity in all the things that positively comprise it as an existing number of parts. Such that each part has a part next to it, excluding none. In this sense one can take up the case of a part that right now has not got a part next to it, and speak of the potential for a going on, as though in the approach of a vanishing horizon.

Aristotle's sense of the infinite, that what is there is not exhaustible, is not wholly clear about what potential means. Is it the case of a kind of Golden Goose, where what comes forth will always pour fourth so long as the actual potential (energia) of the infinity is not struck down in some way? In any case, however much rust and decay has settled on the properly philosophic reflection over this subject matter I do not see where it could be obviously other than an infinite source of fertile attentive investigation.

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Infinity exists and cannot be traversed, completed; infinity is a black hole, but it can be sublimated and transcended. Process (bad infinity) forever, law (real infinity) eternal; process is a variable, law is a constant; Infinity cannot be crossed, but can be transcended, the result of transcendence is a real infinity. The existence of infinity and the imcompletion of process are two completely different concepts, they are two aspects of contradiction and cannot be replaced by each other; it is precisely because of their existence that there is the existence of the contradiction between finity and infinity. Infinity, like a black hole, can come in without going out, endless, never over. The infinite view of dialectical materialism insists on the inextinguishment of this kind of contradiction, and holds that infinity exists objectively and can be recognized, but the infinite process cannot be completed, that is, the infinite contradiction is eternal; On the other hand, the view of actual infinity regards the objective existence of infinity as the completion of the infinite process, replacing the subjective with the objective, replacing the bad infinity with the real infinity, abandoning the contradiction between finity and infinity completely, and thinking that the contradiction can be ended and solved. Therefore, this thought followed Kant's transcendental, subjective, metaphysical infinite thought, not Hegel's dialectical infinite thought. Therefore, to correctly understand and grasp the contradiction between finity and infinity and correctly understand real infinity and bad infinity is the essence of solving the third Mathematical Crisis in the field of mathematics and philosophy, and is of great guiding significance to fully understand the philosophical significance of Russell’s Paradox.

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The so-called Infinite Exchange Paradox refers to the idea that we use the thought of actual infinity-----that is, the idea that the infinite process can be accomplished, and we can transform the two equivalent infinities (with one-to-one correspondence) into mutually nonequivalent infinities. This profoundly exposes the inherent defects of the actual infinite thought, it moves the contradiction to infinity, but the contradiction never disappears. In other words, the infinite process is impossible to complete, thus further supporting the infinite view of dialectical materialism (to be specified in a later chapter).enter image description here

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To sum up, we comprehensively introduce Hegel's dialectical infinity, and also introduce Engels's critical inheritance and development of Hegel's infinite view, it also introduces the differences between the four kinds of infinite views, and analyzes the errors of actual infinity; so now we can naturally sum up the infinite view of dialectical materialism: infinite objective exists, infinite can be known, but infinite process can not be completed.

Concretely speaking, any infinity is the dialectical unity of bad infinity and real infinity. It is an objective existence, and infinity itself contains the contradiction between finity and infinity, so the objective existence of infinity does not mean that the infinite process can end, complete. Real infinity is the inherent qualitative stipulation of infinite things, that is, the inner connection, law and truth, while bad infinity is infinite progress, no terminating repetition and alternating, it deeply embodies the contradiction between finity and infinity.

Real infinity can be recognized and completed, while bad infinity cannot be recognized and can not be completed; bad infinity (infinite process) is the concrete manifestation of the contradiction between finity and infinity rather than the solution of such contradiction; this determines that the contradiction between finity and infinity will never die out. Real infinity represents infinite quality (essence), while bad infinity represents infinite quantity (movement and change).

Real infinity is inseparable from bad infinity, bad infinity is the carrier of real infinity, and real infinity is the goal and direction of bad infinity. The real infinity is present, concrete, positive, rational, completed infinity, is Being-for-self and rational Being, is the completed quality; and the bad infinity is possible, abstract, negative, uncompleted infinite, is Being-in-itself and intellectual Being.

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It is well known that the problem of finity and infinity is the basic problem of mathematics, and it is also the basic problem of Philosophy. The mathematical foundation has experienced three big crises, from the birth of the irrational number, and the rationality of the calculus has been seriously questioned (Hegel, Marx has given a dialectical scientific explanation from the philosophy), and then to the occurrence of Russell Paradox.

The underlying reason behind the three crises is that humanity has not been able to give a scientific and clear explanation to the contradiction between finity and infinity, and there has always been a dispute between actual infinity and potential infinity. The solution of the second crisis is that mathematicians inadvertently follow the idea of dialectics and give the scientific definition of limit. However, since Cantor invented “infinite numbers” (ordinal number, cardinal number) more than 100 years ago, people’s understanding of infinity has once again fallen into the idealistic metaphysics. This paper comprehensively analyzes the essence of actual infinity and retrieves Hegel’s “true essence ofinfinite”, that is, the inner link of infinite—the essence of real infinite, in order to return to the dialectical understanding of infinity in mathematics and philosophy, this is the purpose of this paper.

This paper returns and redefines the infinite view of dialectical materialism, that is, infinity objective existence, infinity can be recognized, infinity cannot be completed. In the philosophical system of dialectical materialism, this paper gives a thorough solution to the famous Continuum Hypothesis problem, and also removes the philosophical obstacle to solve Russell Paradox. Please visit the Philosophy of Mathematics Education Journal, vol. 35 (2019).

http://socialsciences.exeter.ac.uk/education/research/centres/stem/publications/pmej/

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