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Copyright   James R Meyer    2012 - 2024 https://www.jamesrmeyer.com

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Actual, Completed, and Potential Infinity

Page last updated 03 Jul 2023

 

For thousands of years, philosophers have debated the notion that there could be two types of infinity:

  • a “potential” infinity
    and
  • an “actual” infinity (also referred to as a “completed” infinity).

 

A large part of the ambiguity that is inherent in the terminology used in the above paragraph is eliminated by the simple observation that when we use the term “infinity”, what we are really doing is talking about things that are infinite - we use the term ‘infinite’ as an adjective for describing a thing (where the term ‘thing’ could refer to a collection of things, or to a member of a collection). In the same way that we might talk about a thing being red, using the adjective ‘red’ to indicate that it has a property that it preferentially reflects (or emits) light of particular wavelengths, we can talk about something being infinite to indicate that it has the property of being without a limit. For this reason, for the remainder of this page we shall use only terms such as “infinite” rather than the term “infinity”. For example, rewriting our opening paragraph gives us:

 

For thousands of years, philosophers have debated the notion that there could be two types of infinite things:

  • things that are “potentially” infinite
    and
  • things that are “actually” infinite (also referred to as being a “completed” infinite thing).

 

But the expression “potential infinite” is nonsensical, being an oxymoron - either a thing is without any limit, or it is restricted by some sort of limit. To say that a thing is potentially infinite, i.e: potentially without any limit could only mean one of two things:

  1. It is inherently without any limit, in which case, the inclusion of the term potential is superfluous and only serves to confuse, so the term should simply be “infinite”, or
  2. It requires something that changes its properties, so that the thing changes from being something that has a limit to something that does not have a limit, in which case when the term is applied to something, it means, ipso facto, that without such change the thing is not unlimited, and again the term only serves to confuse, and the term in this case should simply be “finite”.

 

The notion of something having the property of being “actually infinite” (actually without any limit) or being “completed infinite” is a similarly incoherent notion. For example, every number in any set of natural numbers is associated back to the natural number zero by a finite number of unitary iterations. If there could be more than a finite number of natural numbers actually simultaneously together in a “completed” set, then there would have to be at least one number that is associated with the natural number zero by more than a finite number of unitary iterations. But that is impossible, since such an entity, by definition, cannot be a natural number.

 

Nevertheless, many mathematicians have continued to insist that infinite things come in these two varieties - and here’s the conundrum that arises from such notions:

The set of all fractions of the form:

110, 1100, 11000, 110000, 1100000 …

is considered by some mathematicians to be a completed infinite set - and the sum of all the elements of this completed set is also said to ‘exist’, but its value is calculated by saying that the series:

110 + 1100 + 11000 + 110000 + 1100000 …

is a potentially infinite series where the limiting value of all sums of a finite number of the terms of the series is given by the limit formula for a geometric series (see Geometric Series) as 19, which is then assumed to be the value of the actual sum that is the completed infinite sum of 110 + 1100 + 11000 + 110000 + 1100000 …

 

Other than finitists who completely reject any mathematics involving the infinite, no mathematician disputes the correctness of that limit formula. The implications of the differing interpretations that can be attached to conundrums such as the above have far-reaching consequences, both in terms of abstract philosophy and also in terms of mathematical foundations.

 

The philosophy of an actual or potential infinite

The origin of the notion of these two different types of infinite things seems to have arisen from the observation that a variable quantity can be such that the value that the variable can be replaced by is always finite, but the domain of the values that it can be replaced by is a set that is infinite - that is, there is no limit to how large the value that can be assigned to the variable, although of course, that value never can actually be infinite. It seems that this has led to the idea that this somehow involves two types of infinite things:

  • an infinite thing that is the “value of the variable that can vary” but which never actually becomes infinite,
    and
  • an infinite thing that is the domain of the variable where the quantity of such values is limitless.

 

However, while one might talk informally about a variable varying in value, the fact is that a variable itself does not have any value, let alone a varying value. As soon as an instance of a variable is replaced by some value from its domain, that instance of that variable can no longer be said to exist in that context. This means that when we informally talk about a “variable varying in value”, what we are really doing is talking about multiple instances of replacements of that variable by different values of its domain. It may be the case that the failure to recognize this subtle distinction has led to a Platonist belief that there actually ‘exist’, in some non-physical sense, variables which can vary in value, and perhaps this has given rise to the notion of two types of infinite things.

 

In classical logic, everything has a fixed value and there is no such thing as an expression with a non-fixed value, only expressions with variables or expressions where a variable has been replaced by some fixed value. (Footnote: More formally, given an expression A with a variable x, then according to the rules of the system, another expression can be derived from that expression by substituting the variable x in that expression A by some member of that variable’s domain.)  (Footnote: I also refer to this fixed value aspect of classical logic at The Drinker paradox where the failure to recognize this aspect of classical logic can result in erroneous conclusions.) For example, in a formal system, an expression such as:

For all x, Some expression (where the expression includes x)

simply implies, by the rules of the system, other expressions where the x in “Some expression” is replaced by some fixed value - the x does not itself have a value.

 

In conventional set theories such as Zermelo-Fraenkel set theory, every entity is a set and there are no entities that are not sets, and this has blurred the subtle distinction between single entities that are finite and collections that are infinite, such as this distinction between a variable and the domain of a variable. (Footnote: In a set theory where elements cannot themselves be sets, there is an inherent delineation between single entities and collections of entities, so that for example, every number is finite while there are infinite sets that have no limit to the quantity of their elements, without any implication that there “exists” any sort of simultaneous “actual” or “completed” collection of those elements. For more on this, see the Overview of set theory.)

 

Let’s now look at one mathematician’s attempt to clarify what he means by potential and completed “infinity” at Eric Schechter’s Potential versus Completed Infinity. He writes:

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, …

gets higher and higher, but it has no end; it never gets to infinity.…

Then he says:

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 …},

With this notation, we are indicating the set of all positive integers. This is just one object, a set. But that set has infinitely many members. By that I don’t mean that it has a large finite number of members and it keeps getting more members. Rather, I mean that it already has infinitely many members.

 

The first case refers to a procedure, an algorithm - but if that implies that each step requires a certain finite amount of time, then that is applying a physical aspect on top of what is ostensibly an entirely abstract concept. We can note that if one were to talk about a ‘procedure’ where all the steps take zero time, then that would imply that the ‘procedure’ completes in zero time - which begs the question: “Would that produce a completed infinity?”

 

The notion that an appeal to a ‘procedure’ can demonstrate that there are two different types of ‘infinity’ involves an inherent confusion of abstract and physical concepts. When we discount any such plea to a commingling of the physical and the abstract, we can see that Schechter’s description fails to clarify anything at all - it provides no explanation whatsoever as to why there is necessarily a one-to-one correspondence between the numbers that constitute this ‘potentially infinite’ thing that “never quite reaches an infinite end” and the numbers that constitute the ‘completed infinite’ thing that supposedly “reaches” “infinity”. We can also observe that such a one-to-one correspondence applies for any other similar infinite sequence.

 

Furthermore, to say that a set “already has infinitely many members” is obviously a Platonist appeal where one is invited to imagine that a definition of a collection where there is no limit to the quantity of entities in the collection either magically conjures up a collection that has some sort of actual non-physical “existence”, or that a corresponding actual non-physical collection conveniently has already ‘existed’ for all time. Schechter’s description amounts to nothing more than vague hand-waving in ambiguous natural language, and it provides no logical rationale for the notion of these two types of infinite things.

 

So, if philosophical attempts at explanations seem to obfuscate rather than elucidate, perhaps there are more formal mathematical definitions that clarify the purported distinction between the so-called potential infinite and the so-called actual infinite? Let’s see.

 

The mathematics of an actual or potential infinite

We can note that Schechter also makes the telling remark that students of mathematics often have difficulty with the notion of completed infinities. But that is hardly surprising when one considers that, in the 140 years or so since Cantor decided that his infinite sets were actual completed infinite sets, there has never been any clear-cut definition of a method that one can use to decide whether an entity is a potentially infinite thing or an actual completed infinite thing. Apparently it comes down to a subjective opinion rather than any objective method. One can peruse web-pages such as The Internet Encyclopedia of Philosophy: The Infinite, the Stanford Encyclopedia of Philosophy: Infinity, and Wikipedia: Actual infinity, but you won’t find any elucidation there. The Wikipedia page states that potential infinity is where “a non-terminating process (such as ‘add 1 to the previous number ’) produces a sequence with no last element, and where each individual result is finite and is achieved in a finite number of steps.” But any infinite set has no last element either, and every individual element is finite, so how might that “definition” distinguish a ‘potentially’ infinite thing and an ‘actual’ infinite thing?

 

The same troubling omission of any definitive distinguishing characteristic between a ‘potential’ infinite and an ‘actual’ infinite is also noted at Stack Exchange: Is there a formal distinction between potential and actual infinities? and Can we formally distinguish between actual and potential infinities? where various notions are put forward, but no-one is able to provide any definitive clarification.

 

Maybe the aforementioned Eric Schechter can provide some clarification? He attempts to justify his notion of a completed infinite set by creating a one-to-one correspondence between the real number points on the real number line and the points of the left half of a partial semi-circle (partial because it excludes the single top and bottom points of the semi-circle). Schechter includes a diagram to illustrate this:

Eric Schechter's diagram of completed infinity
Schechter’s “completed infinite sets”

So, given any straight line going from the center of the partial semi-circle, through any point of the partial semi-circle - excluding the top and bottom points - that line will intersect the real number line at some point. But the same is not true for the top and bottom points of the semi-circle (for these points one would have a straight line parallel to the real number line). (Footnote: Similarly, for any real number, there are infinitely many real numbers on the real number line greater than that number, and for the corresponding point on the partial semi-circle, there are also infinitely many points on the partial semi-circle between that point and the top point of the semi-circle.)

 

Schechter says, that while “there are no points for plus or minus infinity on the line, but it is natural to attach those “numbers” to the endpoints of the semicircle” and he talks about adding + ∞ or - ∞ as a special type of number to either end of the real number line to create what is called the “extended real number line”, and that this is analogous to adding the top and bottom points of the semi-circle to the partial semi-circle to create the entire semi-circle.

 

The plea here is that referring to this extended real number line and the semi-circle somehow implies that there is such a thing as an actual completed infinite set. But Schechter doesn’t even begin to explain how this might in some way imply that there is such a thing as an actual completed infinite set. All he has done is take the infinite set of all real numbers and associated it with the infinite set of points of the partial semi-circle, and then added two elements to the set of real numbers and two elements to the set of points of the partial semi-circle. That does not in any way imply a completed infinite set.

 

One might equally well argue that it implies the opposite - that the set of points of the partial semi-circle is not a completed set - since if it were a completed set actually simultaneously containing all its points, one might expect that there should be a point that is the highest point of that partial semi-circle and which is immediately adjacent to the top point. But, of course, there cannot be such adjacent points, since between any two different real numbers, there is no limit to how many more real numbers one can find between them.

 

In a section on infinite sums, The Internet Encyclopedia of Philosophy: Infinite Sums states the following:

 

Infinite Sums

In the 17th century, when Newton and Leibniz invented calculus, they wondered what the value is of this infinite sum:

11 + 12 + 13 + 14 + …

They believed the sum is 2. Knowing about the dangers of talking about infinity, most later mathematicians hoped to find a technique to avoid using the phrase “infinite sum.” Cauchy and Weierstrass eventually provided this technique two centuries later. They removed any mention of “infinite sum” by using the formal idea of a limit. Informally, the Cauchy-Weierstrass idea is that instead of overtly saying the infinite sum s1 + s1 + s1 + … is some number S, as Newton and Leibniz were saying, one should say that the sequence converges to S just in case the numerical difference between any pair of terms within the sequence is as small as one desires, provided the two terms are sufficiently far out in the sequence. More formally it is expressed this way: The series s1 + s1 + s1 + … converges to S if, and only if, for every positive number ε there exists a number δ such that |sn+h + sn| < ε for all integers n > δ and all integers h > 0. In this way, the reference to an actual infinity has been eliminated.

 

This epsilon-delta technique of talking about limits was due to Cauchy in 1821 and Weierstrass in the period from 1850 to 1871. The two drawbacks to this technique are that (1) it is unintuitive and more complicated than Newton and Leibniz’s intuitive approach that did mention infinite sums, and (2) it is not needed because infinite sums were eventually legitimized by being given a set-theoretic foundation.

 

But if one is choosing one concept over another because it appears less complicated, one needs to be absolutely sure that the simplification is not concealing some important detail, nor introducing an untoward consequence - such as introducing a contradiction. And as for the notion that a limit definition is not needed because “infinite sums were eventually legitimized by being given a set-theoretic foundation”, it can easily be shown that the notion of infinite sums in conventional set theory is contradictory, as is explained in Sums of infinitely many fractions: 1 and Sums of infinitely many fractions: 2 (and other places on this site), such claims of “legitimization” amount to nothing more than looking the other way whilst sweeping uncomfortable facts under the carpet.

 

The only valid conclusion that one can reach is that there has never been any logical argument that proves that things defined as being infinite must be either potentially infinite things or completed actual infinite things. A conclusion that is not surprising since the notion is contradictory. See also the web-page On David Hilbert’s “On the Infinite” which examines Hilbert’s article on infinity and which refers to potential and actual infinity.

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