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.

Hello Bill,

I'm not sure your counter to Peter's modal argument that potential infinity presupposes actual infinity does the argument justice. Here is another way of looking at the question: Suppose, for each n, in possible world n the monks of Hanoi are working their way through an instance of the eponymous tower puzzle with n discs. It seems to me that we can't formulate the notion of possibility in this way without presupposing a completed infinity of possible worlds. There is no sense here that possible world n+1 is the successor of possible world n in some process of counting. The infinitely many worlds are, as it were, conceptually coexistent.

Posted by: David Brightly | Sunday, March 27, 2011 at 07:05 AM

I'd start by commenting on this statement in the main article:

C1. the [natural numbers] do not form a single complete object, the set N, but are such that their infinity is an endless task.

C1 is a novel approach to characterize a set. It describes the natural numbers in terms of supposedly a simpler notion of object. Moreover, C1 implicitly assumes that sets are single complete objects. If I understand C1 correctly, this quite strong assumption should be at least a sufficient condition for an object to be a set.

I’ll first explain what is novel about this approach and then raise a methodological doubt regarding it.

In traditional set theory – meaning set theory not influenced by philosophy of mathematics -- sets are introduced informally as primitive notions. The set exists when there is a well defined or proper membership relation between some objects (set members) and their collection (set). ‘Proper’ nowadays means not leading to Russell’s type of paradoxes relation of membership. Completeness of the object referred to in C1 is not relevant to the question of set’s existence. As long as there is proper membership relation, there is a set. So natural numbers are a well defined set in set theory. So in the context of standard set theory claim #2 in the main entry can be validly inferred from #1, because the membership relation among the members of N is well defined.

My methodological doubt about C1 is this. We have strong reasons to stick to the well working notion of sets in basic set theory. Sets are abstract entities that exist when proper membership relations obtain. Why would we want to introduce much more problematic notions of sets and their existence? Why membership relation is less clear, useful, fruitful than “single complete object”?

Posted by: AR | Sunday, March 27, 2011 at 09:04 AM

To AR: what mathematical or logical terms correspond to the terms 'complete' and 'endless'.

Also, more generally, what are the mathematical or logical translations of 'potential infinity' and 'actual infinity'?

Posted by: Edward Ockham | Sunday, March 27, 2011 at 01:46 PM

Bil,

There are certainly sets of objects that are finite, but in practice not calculable.

For example, the set of all the numbers corresponding to each of the number of hairs in the fur or hair of all the individual mammals on this planet at this instant is a finite set, but non-calculable. It is described by the rule which in theory generates the numbers in the set, as Ed says.

In such cases we might use a "lazy set" which means that when we want a member of the set, we choose a mammal not yet counted, and count its hairs. Repeat when needed.

I think that the notion of a set as a membership relation means that in you can describe the set by a rule which is itself tractable or finite, even if the rule in theory might generate an infinite number of members. In cases where the rule creates potential infinity, one can still use the generating rule as a handle to study the set.

So in the case of your concern that a set contain only actual or potential objects, and not infinite numbers of objects, what about allowing the set to contain either actual countable objects or a rule for making objects, or both?

So the set { 0.3, -1.2, [all natural numbers], [all the cube roots of natural numbers] } has 4 of your countable objects (two regular objects and two rules) even if it is ALSO potentially infinite if actually "flattened" and enumerated?

Posted by: Bill | Sunday, March 27, 2011 at 07:19 PM

MP said... "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."

You are right, there is no such implication. You can deny infinite sets if you like. Some mathematicians do. Their views may be gaining popularity due to the developments in computer science and finite computation.

Posted by: Ethan | Sunday, March 27, 2011 at 09:47 PM

AR,

What I am saying is not novel. But I am presupposing a standard distinction between mathematical and commonsense sets, a distinction you appear not to make.

That sets are treated in set theory as single items 'over and above' their members can be seen from the fact that some sets have sets as members without having their members as members. The power set of {Socrates, Plato} has {Socrates} and {Plato} as members, but it does not have Socrates and Plato as members. Therefore, {Socrates} is distinct from Socrates, and {Plato} from Plato. For if these singletons were identical to their members, then the power set would have Socrates and Plato as members.

Posted by: Bill Vallicella | Tuesday, March 29, 2011 at 05:51 AM

Bill,

Let us distinguish three theses:

(T1) There are sets.

(T2) There are infinite sets.

(T3) There is an actual infinity of natural numbers.

Accepting (T1) does not require accepting (T2) or (T3). One might accept finite sets, but reject both infinite sets as well as an actual infinity of natural numbers. (T2) entails (T1) and anyone who accepts (T2) might as well accept (T3). But (T3) need not entail (T2) because one might accept actual infinity but reject the existence of sets, particularly infinite sets.

If I understand you correctly, your primary target is (T3) and only secondarily (T2). In particular, you wish to maintain that the following two propositions are consistent:

(P1) (T3) is false; i.e., it is not the case that there is an actual infinity of natural numbers;

(P2) "Every natural number has an immediate successor" (i.e., there is no largest natural number).

Your proposal is to view the proposition quoted in (P2) as part of the *sense* of 'natural number'. This proposal, if I understand you correctly, employs Frege's distinction between sense and reference. In line with this distinction, we can think of Thesis (T3) as speaking about the extension of the phrase 'natural number'. So construed, (T3) says that the extension of 'natural number' includes infinitely many natural numbers. How can we resist (T3), yet accept (P2)?

You propose to shift the discussion from the reference, or extension, of 'natural number' to its sense (or intension). So construed, the sense of 'natural number' shall include the sense of 'every natural number has an extension'. Thus, we may purchase (P2), while denying (T3).

But now notice the consequence of this *Fregean Shift*: we ceased speaking of extensions altogether. In fact, following this Freagean Shift, we are not even committed to the existence of one natural number. Therefore, we have not shown that every natural number has a successor is consistent with the denial of an actual infinity, for due to the Fregean Shift we changed the subject: i.e., we now only talk about senses and not extensions.

You need to show the consistency of the following three propositions:

(P3) 'natural number' has an extension;

(P4) The sense of 'natural number' includes 'every natural number has a successor';

(P5) Not-(T3); i.e., it is not the case that there is an actual infinity of natural numbers (or the extension of 'natural number' is not infinite').

I don't think that you have made a cogent case yet for the consistency of these three propositions.

Posted by: Account Deleted | Friday, April 01, 2011 at 05:27 AM

Thanks for taking the time to respond, Peter. I know how busy you are.

>>You need to show the consistency of the following three propositions:

(P3) 'natural number' has an extension;

(P4) The sense of 'natural number' includes 'every natural number has a successor';

(P5) Not-(T3); i.e., it is not the case that there is an actual infinity of natural numbers (or the extension of 'natural number' is not infinite').<<

It seems to me that th three props are consistent. Suppose 'natural number' has the null extension. Then the three are consistent. Now suppose that the extension is non-null. Then it is consistent too if (P4) is interpreted as

P4* The sense of 'natural number' includes 'every natural number has (not actually but potentially) a successor'

I would say you are begging the question aginst me by assuming that 'has' in (P$) means 'actually has.'

It masy be that there is no way to avoid mutual question-begging.

The larger context of this discussion was my claim that no philosophical problem has ever been solved. You disagreed by saying that the philosophical problems concerning infinity had been solved. I honestly don't see that the issue that divides the potentialists from the actualists has been resolved.

Posted by: Bill Vallicella | Friday, April 01, 2011 at 03:13 PM