The Existence of Infinite Sets

A reader asked whether one can  prove that there are actually infinite sets.  Well, let's see.

It occurs to me that 'actually infinite set' is a pleonastic expresson. If there are infinite sets, then they are actually infinite, such that a potentially infinite set would be no set at all. For if there are mathematical (as opposed to commonsense) sets at all, then they are quite definite objects whose identity conditions are supplied by the Axiom of Extensionality: two sets are the same if and only they have all the same members. A mathematical set is not exhausted by its membership -- it is not a mere plurality -- since it is a one to their many; nevertheless, sets are rendered determinate by their members. (Let us for the moment not worry about singletons and the null set which give rise to their own difficulties.) 

It is worth noting that in Georg Cantor's oft-quoted definition, a set (Menge) is a collection of "definite and separate objects." (Contributions to the Founding of the Theory of Tranfinite Numbers, sec. 1) If the members of a set are definite and separate, then the same is true of the set itself. We could say that a math. set inherits its determinacy from the determinacy of its members. 

My point is that, if there are mathematical sets at all, then there is nothing potential, indeterminate, incomplete, or unfinished about them. Each such set is a definite single item distinct from each of its members and from  all of them.  It is a one-over-many. So if there are any infinite sets, then they are actually infinite sets, which is to say that talk of 'actually infinite sets' is redundant.

So our question becomes, Can one prove that there are infinite sets?

I don't know if one can prove it, but one can give an argument. (If a proof is a valid deductive argument the premises of which are self-evident, then damn little can be proven. In particular, the axioms of ZFC are far from self-evident, not that set theorists claim self-evidence for them. Is it self-evident that a null set exists?  Hardly.)

Here is an argument, where 'set' is short for mathematical (as opposed to commonsense) set.

1. There are sets.

2. There are infinitely many natural numbers: no finite cardinal is the number of natural numbers. Therefore,

3. If the natural numbers form a set, then they form an infinite set. (1, 2)

4. The natural numbers form a set.   Therefore,

5. The natural numbers form an infinite set. (3, 4) Therefore,

6. There exists an infinite set. (5)

This is a valid argument, and it renders reasonable its conclusion. But it does not prove its conclusion unless there are proofs for its controversial premises (1) and (4). I argued for (1) in Sets, Pluralities, and the Axiom of Pair.  But what is the argument for (4)?  Why must we think of the natural numbers as forming a set?

Comments

[William]

>>The natural numbers form a set.

I question this. Or rather, I question whether is a plural reference for 'the natural numbers', and I question whether there is an OL set containing every natural number. I agree that every natural number has a successor that is not identical with any predecessor. I don't see that it is intuitive that there is a corresponding OL set, or plurality. Indeed this version of the axiom of infinity

(*) For any OL set S, there is at least one item that is not in S

suggests it is at least logically possible that there is no infinite OL set.

[Bill Vallicella]

Apparently, you did not notice that I am questioning (4), not asserting it. My question is: what arguments are there for (4)? There must be some.

[William]

>>Apparently, you did not notice that I am questioning (4), not asserting it. My question is: what arguments are there for (4)? There must be some.

I did see that, but wanted to make clear that (4) was the difficult one (you mention 1 also, which we agree about).

Are we agreed we are arguing about the existence of an OL infinite set? If it is about M (mathematical) set theory, I have already said that there are versions which deny 4, as well as the standard version which asserts it.

What arguments are there for (4)? Well, if we talk about a domain of things, surely we can always talk about 'the' things, or 'all the things in the domain'. Can't we? But if so it would follow that

(*) For any OL set S, there is at least one item that is not in S

is logically impossible. But I don't see that it is logically impossible. We can meaningfully say that there could be a universe where, for any things whatever in that universe, there is at least one thing that is not one of them. If that is true, there cannot be such things as 'all the things in the universe'. For 'all the things in the universe' falls within the scope of 'any things whatever in that universe'. And if (*) is true, there would be at least one thing that was not one of 'all the things', and so they could not be 'all the things'.

In summary, the only argument for (4) is that it must always be possible to refer to all the things in the domain. But that is only the case if (*) is impossible, and it seems that it is not impossible. (Hope that makes sense).

[William]

(Afterthought). I suppose we could argue as follows. As well as making a plural reference by enumerating the individuals, as in 'Alice and Bob and Chas', we can make an indeterminate reference such as 'the lost ballpoint pens in our house'. I don't know if anyone else has this problem, but our house swallows up these with great rapidity. The only ones you can find by the phone are pencils with broken tips, or pens where the ink has dried up so that you furiously shake it or dip it in water or whatever to take down some important number that your wife's friend wants to leave. There is the possibility that all these lost pens have got swallowed up in some other dimension and are not really in the house, and only exist in a Schrodinger cat kind of way. But we assume not. Despite the fact they never appear again, it seems reasonable to refer to 'the lost pens'.

I reply: that is because we assume there are finitely many lost pens, as clearly they are. That is, we assume that (*) above is false. But that does not mean that it is logically or necessarily false.

Or perhaps we could argue that quantification is like a mental lasso that could rope up absolutely any number of things we like, including an infinite number of things. But (1) that is psychologising and (2) is it necessarily true that a mental lasso could lasso anything? What (*) is asserting is that we cannot lasso all the objects in the domain. We can perhaps lasso a very large finite number, even every finite number of things. But not 'all' things.

Again, hope that makes sense, I have to go.

[William]

Yet another thought. To avoid psychologizing, perhaps we could say that it is an inherent property of a quantifier that it can quantify everything. That is what 'everything' means.

I reply: is that necessarily true? Can 'everything' really get to everything. Note that I am not actually denying the universal scope of the singular quantifier. It is the scope of the plural quantifier 'any things' as in 'take any things that you like' that I am puzzled about. Why should the scope of the plural quantifer include 'all things'?

[Bill Vallicella]

>>I did see that, but wanted to make clear that (4) was the difficult one (you mention 1 also, which we agree about). <<

No, we don't agree about (1), which states that there are sets. I made it clear that 'set' is short for 'mathematical set.' What we agree on is that there are pluralities. A plurality is the referent of a plural expression such as 'the barristers of London.'

We disagree on whether, in addition to pluralities, there are also math. sets.

>>Are we agreed we are arguing about the existence of an OL infinite set?<< No. In the this post I am asking what the reason is for positing infinite mathematical sets such as the math. set of natural numbers.

[William]

>> In the this post I am asking what the reason is for positing infinite mathematical sets such as the math. set of natural numbers.

Well, as I commented, there is no reason one way or the other. If there is no such set, you can still get a theory of number (I don't remember the detail, but I think it's called ZF-inf). If there is, you get the first transfinite set.

I think the question of whether there is an OL inf set is the interesting one.

[David Brightly]

I can offer a reason for wanting to treat the natural numbers as a set. If sets are taken as primitive then the concept of a function from set A to set B can be explained as a subset of the cartesian product AxB, ie, a set of pairs (a,b). To explain in this way the successor function on the natural numbers, say, clearly requires the naturals to be a set. But I don't think this answers the question you are asking.

Here is another distinction between natural language plural terms and mathematical sets. We can prove that the Hatfields form a family. We just have to show that the right relations hold between the people referred to by 'the Hatfields'. Not so with sets. In general, there need be no relations at all between the members of a set. They can be arbitrary pluralities, by stipulation, though we know that we have to tread carefully here to avoid falling into contradiction. So I can't imagine how an argument for showing that the natural numbers form a set might go.

[Bill Vallicella]

David,

You make a good point which I take to be the following. 'The Hatfields' refers to a plurality the members of which stand in certain familiar relations which make of them a family. But the math. set consisting of the Hatfields abstracts from these relations.

'The Hatfields are an Appalachian family' does not commit us ontologically to the existence of an M-set consisting of the members of this family. But 'The Hatfields are 100 in number' does seem to commit us ontologically to an M-set consiting of the members of this family. Why? Because there has to be something distinct from the Hatfields that bears the property of being 100 in number. That is either the M-set consisting of them or the property of being a Hatfield which is 100 times instantiated.

William the nominalist wants to avoid commitment to abstract entities. It seems to me we have to admit some.

'There are four trees in my backyard.' It is not the case that any one of these tress is 4, nor is it the case that the mere plurality is four. There must be a single item over and above the plurality which bears the property of being 4. We could say either:
a) the property of being a tree in my back yard is four times instantiated
or
b) The set {x: x is a tree in my back yard} has four members.

Either way we are committed to an abstract object.

[William]

>>But 'The Hatfields are 100 in number' does seem to commit us ontologically to an M-set consiting of the members of this family. Why? Because there has to be something distinct from the Hatfields that bears the property of being 100 in number.

Why? Does 'A dozen things are 12 in number' commit us to something having the proper of being 12 in number?

>>'There are four trees in my backyard.' It is not the case that any one of these tress is 4, nor is it the case that the mere plurality is four. There must be a single item over and above the plurality which bears the property of being 4.

four(a, b, c, d)

It is not the case that four(a) or four(b) etcetera.

[David Brightly]

Hello Bill,
Yes, that is exactly it. An afterthought: We only need show that the Hatfields form a family if we see them extensionally: Pa Hatfield, Ma Hatfield, and the rest. If we see them intensionally, as that Appalachian family feuding with the McCoys, say, we have nothing to prove. Now, arguably, we cannot grasp the infinity of natual numbers extensionally. We have to see them intensionally as the collection consisting of zero and all its successors. But this concept already has a naive notion of mathematical set built into it. So again we have nothing to prove.

Regarding the trees in your yard, you say 'nor is it the case that the mere plurality is four'. On my understanding of pluralities we are allowed textually to substitute a list of references for the phrase 'a mere plurality' and so arrive at 'nor is it the case that this tree, that tree, the other tree, and the last tree is four'. That must be wrong, surely?

[William]

DB >>Now, arguably, we cannot grasp the infinity of natual numbers extensionally. We have to see them intensionally as the collection consisting of zero and all its successors. But this concept already has a naive notion of mathematical set built into it. So again we have nothing to prove.

But as I have argued, we can easily express the infinity of the natural number series without the concept of an infinite set. Indeed, it is exactly the reverse. We express it by denying its existence.

(*) Any things in the domain are such that there is at least one thing that is not one of them.

I.e. Any things whatever are such that there are a finite number of them.

[Bill Vallicella]

David says, >>Now, arguably, we cannot grasp the infinity of natural numbers extensionally. We have to see them intensionally as the collection consisting of zero and all its successors. But this concept already has a naive notion of mathematical set built into it.<<

William responds: >>But as I have argued, we can easily express the infinity of the natural number series without the concept of an infinite set. Indeed, it is exactly the reverse. We express it by denying its existence.<<

William's response is exceedingly obscure. I don't follow it. But it may be that he is assuming that there is no such thing as actual infinity, that infinity is always only potential. Accordingly, one can always add 1 to a given n to arrive at n + 1. But that is not to say that there is a completed collection of the natural numbers. From 'There is no greatest natural numbers' it does not follow straightaway that there is a math. set of natural numbers.

The language of extension and intension helps. And M-set and its extension are distinct. This is obvious, is it not? Take the singleton {Socrates}. It is distinct from Socrates. I take William to be saying that every set collapses into its extension.

But how is this consistent with the Unrestricted Comprehension Axiom of naive (as opposed to axiomatic) set theory? UCA says that for every condition (property, concept, propositional function) there is a set. If you admit a condition such as 'x shares a domicile with William' then it seems you have to admit that there is a math set {x: x shares a domicile with William}.

[William]

>>William's response is exceedingly obscure. I don't follow it.

It is exceedingly simple and I expressed it formally, without the need for obscure concepts like 'potential infinity'. We assume that at least one set exists in our domain. Then (in mathematical set theory terminology)

(*) Every set in the domain is such that there is at least one element (of the domain) that is not a member of it.

If that is true, and there is at least one set X, then it immediately follows that there is an element that is not in X. Call this e. It then follows from standard set-theoretical assumptions that there is a set X' = X u {e}. The again from (*) above, there is a further element e' that is not in X'. Hence a further set X'', a further element e'' and so on ad infinitum.

But it follows from this that, since every set in the domain excludes at least one element, there is no set of all elements in the domain.

Hence my claim that the infinity of a domain (or series) can be expressed by denying, rather than asserting, the existence of an infinite set. I don't see how this is obscure at all. It is your discussion using obscure concepts like 'potential infinity' and 'extensional' that is obscure.

>>I take William to be saying that every set collapses into its extension.

For OL sets, yes. For M sets, no.

>>But how is this consistent with the Unrestricted Comprehension Axiom of naive (as opposed to axiomatic) set theory? UCA says that for every condition (property, concept, propositional function) there is a set.

Which founders on Russell's paradox. Suppose there is an S such that

S = {x:x not in x}

Then if S is in S, S is not in S. But if S is not in S, S is in S.

[David Brightly]

I take W to be saying that we can axiomatise sets in a way that forbids an infinite set. Such a system may well be adequate for explaining plural terms in natural languages. If so it would seem that infinite sets are in no way 'forced' on us by natural language. But is this adequate for doing mathematics? One of the reasons the mathematician likes his conventional sets in the first place is that they provide a neat way of referencing numbers which share a property, such as being less than ten. Under a system of finite sets the numbers that share the complementary property of being greater than or equal to ten don't form a set and can't be referenced through the language of sets.
My guess is that math sets come about in a different way. Forget for a minute how closely (or not) 'set' and 'member of' used in maths approximate to everyday notions of plurality. Think of set language as completely artificial with terms like 'set' and 'in' whose meaning has to be learned like a foreign language. My claim is that set talk is translatable into talk about conditions. Here are some rather informal examples:
'x in N' means 'x is a natural number'
'x in {2y | y in N}' means 'x is even'
'x in {y | y>10}' means 'x >10'
'x in {y | y>10} intersect {y | y<20}' means '10 < x < 20'
and so on. We can always translate out the set talk. 'N is an infinite set' translates to 'there are infinitely many natural numbers'.

[William]

>>I take W to be saying that we can axiomatise sets in a way that forbids an infinite set.

Correct. This is a standard idea. http://mat140.bham.ac.uk/~wongtl/papers/finitesettheory.pdf

>>If so it would seem that infinite sets are in no way 'forced' on us by natural language. But is this adequate for doing mathematics?

I don't see why not.

>>One of the reasons the mathematician likes his conventional sets in the first place is that they provide a neat way of referencing numbers which share a property, such as being less than ten.

Easily done so long as we can quantify over natural numbers - "for all n such that n < 10 [...]" e.g.

>>Under a system of finite sets the numbers that share the complementary property of being greater than or equal to ten don't form a set and can't be referenced through the language of sets.

We can easily do it by quantification. "for all n such that >= 10 [...]" e.g.

>>We can always translate out the set talk. 'N is an infinite set' translates to 'there are infinitely many natural numbers'.

Not if I am correct. If we allow infinite sets, we get all sorts of other stuff (like the hierarchy, or parts of it). But 'there are infinitely many natural numbers' can be expressed in such a way that there is no hierarchy.

[David Brightly]

>>Easily done so long as we can quantify over natural numbers
>>We can easily do it by quantification.

Certainly. What I'm urging is that set talk is equivalent to quantification talk given a sufficiently rich notion of set, ie, allowing infinities. Further, the set talk is more elegant. Recall the Weierstrassian definition of continuity for a function on the real line: A function f is continuous iff for all a and all e there is a d such that for all x |x-a|<d implies |f(x)-f(a)|<e. Contrast this with the equivalent set talk: A set is open iff each of its points is surrounded by a neighbourhood lying entirely within the set and a function is continuous iff the inverse image of every open set is open.

>> Not if I am correct

If I'm wrong in thinking that we can always translate out the set talk that would suggest that there are results in widely accepted mathematics, where set talk is ubiquitous, that can't be proven if we eschew conventional sets. Can you think of any?

[Jake]

In (2), you have smuggled in the incorrect idea that infinity is a number as opposed to a concept of method. It is true that there is no finite number of natural numbers, but the correct conclusion is that there is *no* number of natural numbers, because the natural numbers cannot be exhaustively enumerated. Infinity indicates the "open-endedness" of certain series; It is not itself a number.

[Bill Vallicella]

Cantor would disagree with you.