On Potential and Actual Infinity, and a Puzzle

Consider the natural numbers (0, 1, 2, 3, . . . n, n +1, . . . ).  If these numbers form a set, call it N, then N will of course be actually infinite.  This because a set in the sense of set theory is a single, definite object, a one-over-many, distinct from each of its members and from all of them.  N must be actually infinite because there is no greatest natural number, and because N contains all the natural numbers. 

It is worth noting that 'actually infinite set' is a pleonastic expression. It suffices to say 'infinite set.'  This is because the phrase 'potentially infinite set' is nonsense. It is nonsense (conceptually incoherent) because a set is a definite object whose definiteness derives from its having exactly the members it has.  A set cannot gain or lose members, and a set cannot have a membership other than the membership it actually has. Add a member to a set and the result is a numerically different set. In the case of the natural numbers, if they form a set, then that set will be an actually infinite set with a definite transfinite cardinality. Georg Cantor refers to that cardinality as aleph-zero or aleph-nought.

I grant, however, that it is not obvious that the natural numbers form a set.  Suppose they don't.  Then the natural number series, though infinite, will be merely potentially infinite.  What 'potentially infinite' means here is that one can go on adding endlessly without ever reaching an upper bound of the series.  No matter how large the number counted up to, one can add 1 to reach a still higher number. The numbers are thus created by the counting, not labeled by the counting.  The numbers are not 'out there' in Plato's topos ouranios waiting to be counted; they are created by the counting.  In that sense, their infinity is merely potential.  But if the naturals are an actual infinity, then  they are not created but labeled.

Moving now from arithmetic to geometry, consider a line segment in a plane.  One can bisect it, i.e., divide or cut it into two smaller segments of equal length.  Thus the segment AB whose end points are A and B splits into the congruent sub-segments AC and CB, where C is the point of bisection. The operation of bisection is indefinitely ('infinitely') iterable in principle.  The term 'in principle' needs a bit of commentary. 

Suppose I am slicing a salami using a state-of-the-art meat slicer. I cannot go on slicing thinner and thinner indefinitely.  The operation of bisecting a salami is not indefinitely iterable in principle.  The operation is iterable only up to a point, and this for the reason that a slice must have a certain minimal thickness T such that if the slice were thinner than T it would no longer be a slice.   But if we consider the space the salami occupies -- assuming that space is something like a container that can be occupied -- then  a longitudinal (non-transversal) line segment running from one end of the salami to the other is bisectable indefinitely in principle.

For each bisecting of a line segment, there is a point of bisection. The question can now be put as follows: Are these points of bisection only potentially infinite, or are they actually infinite?  

A Puzzle

I want to say that from the mere fact that the operation of bisecting a line segment is indefinitely ('infinitely') iterable in principle, it does not follow that the line segment is composed of an actual infinity of points. That is, it is logically consistent to maintain all three of the following:  (i) one can always make  another cut; (ii) the number of actual cuts will always be finite; and that therefore (iii) the number of points in a line will always be finite, and therefore 'infinite' only in the sense that there is no finite cardinal n such that n is the upper bound of the number of cuts. 

At this 'point,' however, I fall into perplexity which, according to Plato, is the characteristic state of the philosopher. If one can always make another cut, then the number of possible cuts cannot be finite. For if the number of possible cuts is finite, then it can longer be said that the line segment has a potentially infinite number of points of bisection.  It seems that a potential infinity of actual cuts logically requires an actual infinity of possible cuts.

But then actual infinity, kicked out the front door, returns through the back door.

I have just posed a problem for those who are friends of the potentially infinite but foes of the actual infinite. How might they respond?

 

 

 

 

 

 

Comments

[Brian Bosse]

Hello Bill,

I am a friend of the potentially infinite.

It seems to me that your bisection problem can be applied mutatis mutandis to the generation of the natural numbers via the successor function. Consider the following formulation of your puzzle.

If ‘0’ is a natural number, and one can always apply the successor function ‘S(n)’ to get a new natural number ‘n+1’, then the number of possible applications of the successor function cannot be finite. For if the number of possible applications of the successor function is finite, then it can no longer be said that the natural numbers are potentially infinite. It seems that a potential infinity of numbers requires an actual infinity of possible applications of the successor function.

Is this the same problem? If so, then how one responds to this would seem to apply to your bisection problem. Consider the following inductive definition for the series of natural numbers (NN):

1. There is a first ‘thing’, say, ‘0’ that is a natural number.
2. If n is a ‘thing’, then Sn is a ‘thing’.
3. There is nothing else that is a ‘thing’.

From this we can define ‘0’ as ‘0’, ‘S0’ as ‘1’, ‘S00’ as ‘2’, etc… and call this series the natural numbers (NN). NN is said to be potentially infinite because there is no upper bound to the series.

Let’s grant that the number of possible applications of the successor function is not finite. It does not seem obvious to me that this entails the actual existence of a set of elements that are “possible applications of the successor function.” In addition to this, it does not seem obvious to me that a “possible application of the successor function” is something that could be an element of a set. So, why assume that these possible applications make up a set that is actually infinite? Why not say that in the same way NN is potentially infinite, the application of the successor function in the construction of NN is potentially infinite?

Brian

[Christopher McCartney]

In the statement, "If one can always make another cut, then the number of possible cuts cannot be finite," the phrase 'the number of possible cuts' embodies a hidden assumption: that there is some number which is the total number of possible cuts. Given this assumption, you very reasonably note that no finite number fits the bill, from which it follows that x would have to be an infinite number. But what if there is no such number in the first place? Perhaps it is not possible to answer the question "How many possible cuts are there?" by giving a single number which is exactly that total. Perhaps the only way to answer the "how many" question is by saying which numbers are such that there are at least that many possible cuts (the answer being: all of them).

Someone might even say the fact that "one can always make another cut" is a good reason to think there is no total number of possible cuts. For suppose there were such a number w; then after making w cuts one could not make any more cuts, contrary to the hypothesis that one can always make another cut.

Not that this would convince the believer in actual infinities, who can interpret the "always" as meaning not "after any number of cuts" but "after any finite number of cuts." Still, I think it is sufficient to show that skepticism about the existence of the number in question is not unmotivated. Even the proponent of actual infinities will have to grant that in some situations when a "how many" question initially seems to make sense, there is no such number: How many transfinite numbers are there?

[Michael Brazier]

Brian, to make that argument work you must think of a function as a process that generates a number from another number, not (as mathematicians normally do) an association between two sets of numbers. The difficulty with that is, the "generation" in question is timeless. We didn't produce 1 by applying the process of "succession" to 0 - 0 and 1 simply exist as abstractions, and 1 stands in the relation of "successor" to 0. If a function can be applied to a number, it has been applied, so the result of that function has as much existence as its argument does. There's no such thing as an abstract object that's merely possible.

A better solution to the puzzle would be to say that infinite sets exist as abstractions, but can never be fully realized in material things. For, of course, any physical process like adding one more grain to a pile of sand or cutting one more slice of salami takes time, and therefore cannot be iterated indefinitely. But we can contemplate, and reason about, what would happen if such a process were iterated indefinitely, and if we're careful we will reach valid conclusions when we do so - as Isaac Newton did by developing calculus and using it to gain insight into local motion and gravity, for example.

[Joe Odegaard]

The "Meat Slicer" Moto Guzzi, State of the art, 1931. Enjoy. This is from the realm of the non-political, not enjoyable by leftists.

https://www.youtube.com/watch?v=C0aIxOXzVRk

[BV]

Hi Brian,

I was hoping to draw you out, and I'm glad I did. (My nemesis Buckner is who I really had in mind, though.) Your comment is a good one.

I think that there is one problem here that can be approached in two different but related ways, arithmetically and geometrically. It is telling that when we try to think concretely and precisely about numbers (positive and negative integers, fractions, irrational numbers, complex (imaginary) numbers) we are forced to use diagrams. And we are forced to use spatial words, e.g., 'between any two rational numbers there is at least one irrational number.'

>>Why not say that in the same way NN is potentially infinite, the application of the successor function in the construction of NN is potentially infinite?<<

I grant you that the ACTUAL addings of 1 to any given nat'l number is only potentially infinite. But whatever is actual must be possible. So if there is no upper bound to the natural number series, and thus no upper bound to the number of addings, then the number of POSSIBLE addings of 1 cannot be merely potential but must be actually infinite.

Note that I spoke of an operation (operator-operand-output) whereas you speak of a function. Now a function f(x) is a relation between two sets, a domain and a range such for every x in the domain there is a unique y in the range. Thus squaring is function because it maps both 2 and -2 onto a unique y, namely, 4. Successor is obviously a function in that it maps 0 onto 1, 1 onto 2, 2 onto 3, and so on. Your talk of functions does not sit well with your belief that infinity is potential not actual. For if every nat'l number has a successor, and a function is a relation between sets, then the range of the successor function must be an infinite set. I proved that talk of a potentially infinite set is incoherent.

This is not as clear as I would like it to be. I'll try again tomorrow.

[oz]

Sorry for not replying earlier. I am still fretting about the spatiality of sensation.

>If one can always make another cut, then the number of possible cuts cannot be finite.

Nominalists say that the conception of an actual infinity of natural numbers depends on there being a set of all such numbers. But Ockhamists do not believe in sets. They say that the term ‘a pair of shoes’ is a collective noun which deceives by the singular expression ‘a pair’. Deceives, because it means no more than ‘two shoes’, and if there is only a pair of shoes, then there are only two things. But if a ‘pair’ of two things is a single thing, there are three things, the two things and the pair. Ergo etc.

Ockhamists then say that we can do away with sets by plural quantifiers, and thus construct natural numbers without bogus entities. Take any possible number of cuts, that is, any cuts (thus plurally quantifying). Suppose that any possible cuts P are such that there is at least one further cut, thus P+1 cuts. But P+1 cuts are also covered by the quantifier ‘any cuts’. Ergo P+1+1 cuts, ergo ergo ergo ….

What about 'all' possible cuts, you ask? I reply, 'all' is covered by the quantifier 'any'. Therefore (reductio) if all possible cuts P* exist, then there is yet another possible cut, giving P*+1 cuts, in which case P* are not all possible cuts. So there are no P* such that they are all possible cuts.

[oz]

> if ... a function is a relation between sets

See my post just now. Can we have functions without sets? I believe so.