The Big Henry offers the following comment on my post, World + God = God?

"World + God = God" is (mathematically) analogous to "number + infinity = infinity", where "number" is finite. If God embodies all existence, then God is "existential infinity", and, therefore, no amount of existence can be added to or subtracted from God's totality.The numerical concept of infinity does not comply with the rules of arithmetic addition or subtraction. Similarly, if God is presumed to be the embodiment of all existence, He does not comply with the rules of arithmetic addition or subtraction.

To supply an example that supports Big Henry's point, 8 + = . (aleph-nought, aleph-zero, aleph-null) is the first transfinite cardinal. A cardinal number answers the How many? question. Thus the cardinal number of the set {Manny, Moe, Jack} is 3, and the cardinal number of {1, 3, 5, 7} is 4. Cardinality is a measure of a set's size. What about the infinite set of natural numbers {0, 1, 2, 3, 4 . . . n, n + 1, . . .}? How many? . And as was known long before Georg Cantor, it is possible to have two infinite sets, call them E and N such that E is a *proper* subset of N, but both E and N have the same size or cardinality. Thus the evens are a proper subset of the naturals, but there are just as many of the former as there are of the latter, namely, . How can this be? Well, EACH element of the evens can be put into 1-1 correspondence with an element of the naturals.

So far the analogy holds. But I think Big Henry has overlooked the transfinite *ordinals*. The first transfinite ordinal, denoted , is the order type of the set of nonnegative integers. (See here.) You could think of as the successor of the natural numbers. It is the first number following the entire infinite sequence of natural numbers. (Dauben, 97) The successor of is + 1. These two numbers are therefore different. Here the analogy breaks down. God + Socrates = God. + 1 is not equal to .

Moreover, it is not true to say that "The numerical concept of infinity does not comply with the rules of arithmetic addition or subtraction." This ignores the rules of transfinite cardinal arithmetic and those of transfinite ordinal arithmetic. Big Henry seems to be operating with a pre-Cantorian notion of infinity. Since Cantor we have an exact mathematics of infinity.

In any case, I rather doubt that mathematical infinity provides a good analogy for the divine infinity. God is not a set!

Returning to a discussion we were having back in August of 2010, I want to see if I can get Peter Lupu to agree with me on one point: *It is not obvious or compellingly arguable (arguable in a 'knock-down' way) that there are infinite sets.* Given my aporetic concerns, which Peter thoroughly understands, I will be satisfied if I can convince him that the italicized sentence is true, and therefore that the thesis that the infinite in mathematics is potential only is respectable and defensible and has never been shown definitively to be false. Let us start with a datanic claim that no one can reasonably deny:

1. There are infinitely many natural numbers.

If anyone were to deny (1) I would show him the door. For anyone who denied (1) would show by his denial that he did not grasp the sense of 'natural number.' The question, however, is whether from (1) we can validly infer

2. There is a *set* of natural numbers.

If there is such a set, then of course it is an infinite set, an actually infinite set. (Talk of potentially infinite sets is nonsense as I have argued in previous posts.) So, if the inference from (1) to (2) is valid, we have a knock-down proof of actual infinity. For if there are infinite sets then there are actual infinities, completed infinities.

Now I claim that it is obvious that (2) does not follow from (1). For it might be that the naturals do not form a set. A set is a one-over-many, a definite single object distinct from each of its members and from all of them. It should be obvious, then, that from the fact that there ARE many Fs it does not straightaway follow that there IS a single thing comprising these many Fs. This is especially clear in the case of infinitely many Fs.

But from Logic 101 we know that an invalid argument can have a true conclusion. So, despite the fact that (2) does not follow from (1), it might still be the case that (2) is true. I might be challenged to say what (1) could mean if it does not entail (2). Well, I can say that however many numbers we have counted, we can count more. If we have counted up to *n*, we can add 1 and arrive at *n *+ 1. The procedure is obviously indefinitely iterable. That means: there is no definite *n* such one can perform the procedure only *n* times. One can perform it indefinitely many times. Accordingly, 'infinitely many' behaves differently than 'finitely many.' If something can be done only finitely many times, then there is some finite *n* such that *n* is the number of times the thing can be done. But 'infinitely many' does not require us to say that that there is some definite transfinite cardinal which is the number of times a thing that can be done infinitely many times can be done. For 'infinitely many' can be construed to mean: indefinitely many.

On this approach, the naturals do not form a single complete object, the set N, but are such that their infinity is an endless task. The German language allows a cute way of putting this: *Die Zahlen sind nicht gegeben, sondern aufgegeben.* In Aristotelian terms, the infinity of the naturals is potential not actual. But if you find these words confusing, as Peter does, they can be avoided. A wise man never gets hung up on words.

Now if I understood him aright, one of Peter's objections is that the approach I am sketching implies that there is a last number, one than which there is no greater. But it has no such implication. For the very sense of 'natural number' rules out there being a last number, and this sense is understood by all parties to the dispute. There cannot be a last number precisely because of the very meaning of 'number.' *Every* natural number is such that it has an immediate successor. But from this it does not follow that there is a *set* of natural numbers. For 'has an immediate successor' needn't be taken to mean that each number has now a successor; it can be taken to mean that each number at which we have arrived by computation is such that an immediate successor can be computed by adding 1.

But Peter has a stronger objection, one that I admit has force. His objection *in nuce* is that potential infinity presupposes actual infinity. Peter points out that my explanation of what it means to say that the naturals are potentially infinite makes use of words like 'can.' Thus above I said, "however many numbers we have counted, we *can* count more." This 'can' refers either to the abilities of men or machines or else it refers to abstract possibilities of counting not tied to the powers of men or machines.

Consider the second idea, the more challenging of the two. Suppose the universe ceases to exist at a time t right after some huge but finite n has been computed. Now n cannot be the last number for the simple reason that there cannot be a last number. This 'cannot' is grounded in the very sense of 'natural number.' So it must be possible that 1 be added to n to generate its successor. And it must be possible that 1 be added to n + 1 to generate *its *successor, and so on. So Peter could say to me, "Look, you have gotten rid of an actual infinity of numbers but at the expense of introducing an actual infinity of unrealized possibilities of adding 1: the possibility P1 of adding 1 to n; the possibility P2 of adding 1 to n + 1, etc."

The objection is not compelling. For I can maintain that the unrealized possibilities P1, P2, . . . Pn, . . . all 'telescope,' i.e., collapse into one generic possibility of adding 1. P1 is the possibility of adding 1 to n and P2 is the possibility of adding 1 to the last number computed just before the universe ceases to exist.

What I'm proposing is that 'Every natural number has an immediate successor' is true solely in virtue of the sense or meaning of 'natural number.' Its being true does not require that there be, stored up in Plato's Heaven, a completed actual infinity of naturals, a* set *of same. Since I have decidedly Platonic sympathies, I would welcome a refutation of this proposal.

A reader writes,

Regarding your post about Cantor, Morris Kline, and potentially vs. actually infinite sets: I was a math major in college, so I do know a little about math (unlike philosophy where I'm a rank newbie);

on the other hand, I didn't pursue math beyond my bachelor's degree so I don't claim to be an expert. However, I do know that we never used the terms "potentially infinite" vs. "actually infinite".

I am not surprised, but this indicates a problem with the way mathematics is taught: it is often taught in a manner that is both ahistorical and unphilosophical. If one does not have at least a rough idea of the development of thought about infinity from Aristotle on, one cannot properly appreciate the seminal contribution of Georg Cantor (1845-1918), the creator of transfinite set theory. Cantor sought to achieve an exact mathematics of the actually infinite. But one cannot possibly understand the import of this project if one is unfamiliar with the distinction between potential and actual infinity and the controversies surrounding it. As it seems to me, a proper mathematical education at the college level must include:

1. Some serious attention to the history of the subject.

2. Some study of primary texts such as Euclid's *Elements*, David Hilbert's *Foundations of Geometry*, Richard Dedekind's *Continuity and Irrational Numbers*, Cantor's *Contributions to the Founding of the Theory of Transfinite Numbers,* etc. Ideally, these would be studied in their original languages!

3. Some serious attention to the philosophical issues and controversies swirling around fundamental concepts such as set, limit, function, continuity, mathematical induction, etc. Textbooks give the wrong impression: that there is more agreement than there is; that mathematical ideas spring forth ahistorically; that there is only one way of doing things (e.g., only one way of construction the naturals from sets); that all mathematicians agree.

Not that the foregoing ought to *supplant* a textbook-driven approach, but that the latter ought to be *supplemented* by the foregoing. I am not advocating a 'Great Books' approach to mathematical study.

Given what I know of Cantor's work, is it possible that by "potentially infinite" Kline means "countably infinite", i.e., 1 to 1 with the natural numbers?

No!

Such sets include the whole numbers and the rational numbers, all of which are "extensible" in the sense that you can put them into a 1 to 1 correspondence with the natural numbers; and given the Nth member, you can generate the N+1st member. The size of all such sets is the transfinite number "aleph null". The set of all real numbers, which includes the rationals and the irrationals, constitute a larger infinity denoted by the transfinite number C; it cannot be put into a 1 to 1 correspondence with the natural numbers, and hence is not generable in the same way as the rational numbers. This would seem to correspond to what Kline calls "actually infinite".

It is clear that you understand some of the basic ideas of transfinite set theory, but what you don't understand is that the distinction between the countably (denumerably) infinite and the uncountably (nondenumerably) infinite falls on the side of the actual infinite. The countably infinite has nothing to do with the potentially infinite. I suspect that you don't know this because your teachers taught you math in an ahistorical manner out of boring textbooks with no presentation of the philosophical issues surrounding the concept of infinity. In so doing they took a lot of the excitement and wonder out of it. So what did you learn? You learned how to solve problems and pass tests. But how much actual understanding did you come away with?

Morris Kline, *Mathematics: The Loss of Certainty*, Oxford 1980, p. 200:

. . . when Cantor introduced actually infinite sets, he had to advance his creation against conceptions held by the greatest mathematicians of the past. He argued that the potentially infinite in fact depends on a logically prior actually infinite. He also gave the argument that the irrational numbers, such as the square root of 2, when expressed as decimals involved actually infinite sets because any finite decimal could only be an approximation.

Here may be one answer to the question that got me going on this series of posts. The question was whether one could prove the existence of actually infinite sets. Note, however, that Kline's talk of actually infinite sets is pleonastic since an infinite set cannot be anything other than actually infinite as I have already explained more than once. Pleonasm, however, is but a peccadillo. But let me explain it once more. A potentially infinite set would be a set whose membership is finite but subject to increase. But by the Axiom of Extensionality, a set is determined by its membership: two sets are the same iff their members are the same. It follows that a set cannot gain or lose members. Since no set can increase its membership, while a potentially infinite totality can, it follows that that there are no potentially infinite sets. Kline therefore blunders when he writes,

However, most mathematicians -- Galileo, Leibniz, Cauchy, Gauss, and others -- were clear about the the distinction between a potentially infinite set and an actually infinite one and rejected consideration of the latter. (p. 220)

Kline is being sloppy in his use of 'set.' Now to the main point. Suppose you have a right triangle. If two of the sides are one unit in length, then, by the theorem of Pythagoras, the length of the hypotenuse is the sqr rt of 2 = 1.4142136. . . . Despite the nonterminating decimal expansion, the length of the hypotenuse is perfectly definite, perfectly determinate. If the points in the line segment that constitutes the hypotenuse did not form an infinite set, then how could the length of the hypotenuse be perfectly definite? This is not an argument, of course, but a gesture in the direction of a possible argument.

If someone can put the argument rigorously, have at it.

In an earlier thread David Brightly states that "The position on potential infinity that he [BV] is defending is equivalent to the denial of the principle of mathematical induction." Well, let's see.

1. To avoid lupine controversy over 'potential' and 'actual,' let us see if we can avoid these words. And to keep it simple, let's confine ourselves to the natural numbers (0 plus the positive integers). The issue is whether or not the naturals form a set. I hope it is clear that if the naturals form a set, that set will not have a finite cardinality! Were someone to claim that there are 463 natural numbers, he would not be mistaken so much as completely clueless as to the very sense of 'natural number.' But from the fact that there is no finite number which is the number of natural numbers, it does not follow that there is a *set* of natural numbers.

2. So the dispute is between the Platonists -- to give them a name -- who claim that the naturals form a set and the Aristotelians -- to give them a name -- who claim that the naturals do not form a set. Both hold of course that the naturals are in some sense infinite since both deny that the number of naturals is finite. But whereas the Platonists claim that the infinity of naturals is completed, the Aristotelians claim that it is incomplete. To put it another way, the Platonists -- good Cantorians that they are -- claim that the naturals, though infinite, are a definite totality whereas the Aristoteleans claim that the naturals are infinite in the sense of indefinite. The Platonists are claiming that there are definite infinities, finite infinities -- which has an oxymoronic ring to it. The Aristotelians stick closer to ordinary language. To illustrate, consider the odds and evens. For the Platonists, they are infinite disjoint subsets of the naturals. Their being disjoint from each other and non-identical to their superset shows that for the Platonists there are definite infinities.

3. Suppose 0 has a property P. Suppose further that if some arbitrary natural number n has P, then n + 1 has P. From these two premises one concludes by *mathematical induction* that all n have P. For example, we know that 0 has a successor, and we know that if arbitrary n has a successor, then n +1 has a successor. From these premises we conclude by mathematical induction that all n have a successor.

4. Brightly claims in effect that to champion the Aristotelean position is to deny mathematical induction. But I don't see it. Note that 'all' can be taken either distributively or collectively. It is entirely natural to read 'all n have a successor' as 'each n has a successor' or 'any n has a successor.' These distributivist readings do not commit us to the existence of a set of naturals. Thus we needn't take 'all n have a successor' to mean that the set of naturals is such that each member of it has a successor.

5. Brightly writes, "My understanding of 'there is' and 'for all' requires a pre-existing domain of objects, which is why, perhaps, I have to think of the natural numbers as forming a set." Suppose that the human race will never come to an end. Then we can say, truly, 'For every generation, there will be a successor generation.' But it doesn't follow that there is a set of all these generations, most of which have not yet come into existence. Now if, in this example, the universal quantification does not require an actually infinite set as its domain, why is there a need for an actually infinite set as the domain for the universal quantification, 'Each n has a successor'?

6. When we say that each human generation has a successor, we do not mean that each generation *now* has a successor; so why must we mean by 'every n has a successor' that each n *now* has a successor? We could mean that each n is such that a successor for it can be constructed or computed. And wouldn't that be enough to justify mathematical induction?

**Addendum 8/15/2010 11:45 AM.** I see that I forgot to activate Comments before posting last night. They are on now.

It occurred to me this morning that I might be able to turn the tables on Brightly by arguing that actual infinity poses a problem for mathematical induction. If the naturals are actually infinite, then each of them enjoys a splendid Platonic preexistence vis-a-vis our computational activities. They are all 'out there' in Plato's heaven/Cantor's paradise. Now consider some stretch of the natural number series so far out that it will never be reached by any computational process before the Big Crunch or the Gnab Gib, or whatever brings the whole shootin' match crashing down. How do we know that the naturals don't get crazy way out there? How can we be sure that the inductive conclusion *For all n, P(n)* holds? *Ex hypothesi*, no constructive procedure can reach out that far. So if the numbers exist out there, but we cannot reach them by computation, how do we know they behave themselves, i.e. behave as they behave closer to home? This won't be a problem for the constructivist, but it appears to be a problem for the Platonist.

This is wild stuff; I cannot say whether it is mathematically respectable but the man does teach at Rutgers. It is certainly not mainstream. Excerpt:

It is utter nonsense to say that sqrt 2

isNow what the hell does that mean? A rational number is one that can be expressed as a fraction a/b where both a and b are integers and b is not 0. An irrational number is one that cannot be expressed in this way. By the celebrated theorem of Pythagoras, a right triangle with sides of 1 unit in length will have an hypotenuse with length = the square root of 2. This is an irrational number. But this irrational number measures a quite definite length both in the physical world and in the ideal world. How can this number not exist? It is inept to speak of a symbol as shorthand for an ideal object since, if x is shorthand for y, then both are linguistic items. For example 'POTUS' is shorthand for 'president of the United States.' But 'POTUS' is not shorthand for Obama. 'POTUS' refers to Obama. Zeilberger appears to be falling into use/mention confusion. If the symbol for the sqrt of 2 refers to an ideal object, then said object is a number that does exist. And in that case Zeilberger is contradicting himself.

What's more, it seems that from Zeilberger's own example one can squeeze out an argument for actual infinity. We note first that the decimal expansion of the the sqrt of 2 is nonterminating: 1.4142136 . . . . We note second that the length of the hypotenuse is quite definite and determinate. This seems to suggest that the decimal expansion must be actually infinite. Otherwise, how could the length of the hypotenuse be definite?

As an ultrafinitist, however, Zeilberger denies both actual *and potential* infinity:

. . . the philosophy that I am advocating here is called

So I deny even the existence of the Peano axiom that every integer has a successor.

As I said, this is wild stuff. He may be competent as a mathematician; I am not competent to pronounce upon that question. But he appears to be an inept philosopher of mathematics. But this is not surprising. It is not unusual for competent scientists and mathematicians to be incapable of talking coherently about what they are doing when they pursue their subjects. Poking around his website, I find more ranting and raving than serious argument.

The ComBox is open if someone can clue us into the mysteries of ultrafinitism. There is also some finitist Russian cat, a Soviet dissident to boot, name of Esenin-Volpin, who Michael Dummett refers to in his essay on Wang's Paradox, but Dummett provides no reference. Is ultrafinitism the same as strict finitism?