I posed the question, Can one prove that there are infinite sets? Researching this question, I consulted the text I studied when I took a course in set theory in a mathematics department quite a few years ago. The text is Karl Hrbacek and Thomas Jech, *Introduction to Set Theory* (Marcel Dekker, 1978). On pp. 53-54 we read:

It is useful to formulate Theorem 2.4 a little differently. We call a set A inductive if (a) 0 is an element of A; (b) if x is an element of A, then S(x) is an element of A. [The successor of a set x is the set S(x) = x U {x}.]

In this terminology, Theorem 2. 4 is asserting that the set of natural numbers is inductive. There is only one difficulty with this reformulation: We have not yet proved that the set of all natural numbers exists. There is a good reason for it: It cannot be done, axioms adopted so far do not imply existence of infinite sets. Yet the possibility of collecting infinitely many objects into a single entity is the essence of set theory and the main reason for its usefulness in many branches of abstract mathematics. We, therefore, extend our axiomatic system by adding to it the following axiom.

The Axiom of Infinity. An inductive set exists.

Intuitively, the set of all natural numbers is such a set.

Therefore, if we turn to the mathematicians for help in answering our question, we get the following. There are infinite (inductive) sets because we simply posit their existence! Thus their existence is not proven, but simply assumed. Philosophically, this leaves something to be desired. For it is not self-evident that there should be any infinite sets. If there are infinite sets, then they are actually, not potentially, infinite. (The notion of a potentially infinite mathematical set is senseless.) And it is not self-evident that there are actual infinities.

I will be told that there is no necessity that an axiom be self-evident. True: axiomhood does not require self-evidence. But if an axiom is an arbitrary posit, then I am free to reject it. Being a cantankerous philosopher, however, I demand a bit more from a decent axiom. I suppose what I am hankering after is a compelling reason to accept the Axiom of Infinity.

A comparison with complex (imaginary) numbers occurs to me. They are strange animals. But however strange they are, there is a sort of argument for them in the fact that they 'work,' i.e. they find application in alternating current theory the implementation of which is in devices all around us. But can a similar argument be made for the denizens of Cantor's Paradise? I don't know, but I have my doubts. Nature is finite and so not countably infinite let alone uncountably infinite. But *caveat lector*: I am not a philosopher of mathematics; I merely play one in the blogosphere. What you read here are jottings in an online notebook. So read critically.

>>Therefore, if we turn to the mathematicians for help in answering our question, we get the following. There are infinite (inductive) sets because we simply posit their existence!

That's why they are axioms.

>>Thus their existence is not proven, but simply assumed.

Yes.

>>Philosophically, this leaves something to be desired.

Yes.

>>For it is not self-evident that there should be any infinite sets.

Yes.

>>If there are infinite sets, then they are actually, not potentially, infinite.

Also yes.

>>The notion of a potentially infinite mathematical set is senseless.

Agreed.

>> And it is not self-evident that there are actual infinities.

OK.

>>A comparison with complex (imaginary) numbers occurs to me. They are strange animals. But however strange they are, there is a sort of argument for them in the fact that they 'work,' i.e. they find application in alternating current theory the implementation of which is in devices all around us. But can a similar argument be made for the denizens of Cantor's Paradise? I don't know, but I have my doubts.

I too.

So, two philosophers in absolute agreement, today.

Posted by: William | Friday, July 23, 2010 at 02:06 PM

Holy moly! Break out the champagne!

Posted by: Bill Vallicella | Friday, July 23, 2010 at 05:51 PM

I hate to break up this joyful party that Bill and William seem to be enjoying, but I fail to see what are they celebrating. William denies that there are sets altogether, finite or infinite. Bill thinks there are sets, finite as well as infinite.

As to the question about the existence of infinite sets in mathematics (e.g., the set of all natural numbers). The question here is not about the existence of sets as such, but about the existence of sets with infinite cardinality. And while the existence of such sets is stipulated as an axiom and cannot be proven from other axioms, the more fundamental question is what are the consequences and the cost of denying this axiom. For instance, if one denies the existence of an infinite set of natural numbers, but accepts the existence of finite sets, then I challenge such a person to answer the following question: what is the largest member of the set of all natural numbers?

Posted by: Account Deleted | Friday, July 23, 2010 at 09:12 PM

>>William denies that there are sets altogether, finite or infinite. Bill thinks there are sets, finite as well as infinite.

I do not deny there are 'OL sets' (ordinary language sets). These are identical with their extension. We also number them differently. E.g. Two things are two things, but one pair of things is one pair, not two. We avoid paradox (I argue) because one pair is not one thing. So I don't deny the existence of sets, construed in this way.

>>As to the question about the existence of infinite sets in mathematics (e.g., the set of all natural numbers). The question here is not about the existence of sets as such, but about the existence of sets with infinite cardinality.

This leads to the interesting question of whether there are infinite OL sets. If it is a given that there is at least one number, and that every number has a successor non-identical with any predecessor, so there is no 'last number' - then is there an infinite OL set or not? If every OL set is identical with its extension, it would seem to follow that there is an infinite OL set. But there are problems with this, as I have already suggested.

>> And while the existence of such sets is stipulated as an axiom and cannot be proven from other axioms, the more fundamental question is what are the consequences and the cost of denying this axiom. For instance, if one denies the existence of an infinite set of natural numbers, but accepts the existence of finite sets, then I challenge such a person to answer the following question: what is the largest member of the set of all natural numbers?

Denying the existence of an infinite set (mathematical set) is perfectly consistent with there being infinitely many natural numbers. This is known as 'ZF-inf' - Zermelo Fraenkel set theory with the axiom of infinity negated. See http://mat140.bham.ac.uk/~wongtl/papers/finitesettheory.pdf for a discussion of this.

In answer to your question about the largest natural number, there is none. Indeed this follows from the key assumption of ZF-inf

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

It follows both that there is no infinite set, and that there is no 'last element'. However high you go, there is always one element left over. You can combine this with the set you just reached, to form a larger set, and then there is another element not in that, and so on

ad infinitum. Thus, no infinite set, and no 'last element'.This is elementary. The really interesting question, as I say, is whether the same is true in ordinary languate set theory. There are no axioms in OL set theory, only definitions. Everything follows logically. Then the question: is Axinf in OL true or false?

Posted by: William | Friday, July 23, 2010 at 11:52 PM

Are we to take it as either self-evident or apodictically proven that nature is finite? I understand that standard versions of contemporary cosmology holds that to be the case. But, for all its sophisticated mathematics, does not contemporary cosmology still rests upon an inductive basis?

Posted by: Richard E. Hennessey | Saturday, July 24, 2010 at 04:49 AM

William,

Just a few comments.

First, I have no clue what are 'OL-sets' (i.e., ordinary language sets). About these OL-sets you say:

"These are identical with their extension. We also number them differently. E.g. Two things are two things, but one pair of things is one pair, not two. We avoid paradox (I argue) because one pair is not one thing. So I don't deny the existence of sets, construed in this way."

What does the phrase 'their extension' refers to here? If by 'their extension' you mean the collection of items that form an OL-set, then I am unsure in what sense do we have here something that should be called a set, OL or otherwise. We simply have the members, that is it. On the other hand, you maintain that a pair is one pair but is not one thing. That seems to me obfuscating. Why isn't a pair a thing? Does a pair of shoes exists? If so, why isn't it a 'thing' just like one shoe is a thing. After all a shoe is just a collection of a number of things put together (a sole, and so forth).

Second, I cannot comment on the question of whether there is an infinite OL-set until I grasp what is an OL-set.

Third, as for the ZF-inf matter. I *suspect* (although I cannot prove this) that the metalanguage of the proof presupposes the existence of an infinite set. Since the proof is model theoretic, I cannot imagine how the metalanguage in which the proof is given does not itself have an infinite model; how else could they even express ZF + the infinite axiom unless there is a model which satisfies this axiom in the metalanguage.

Posted by: Account Deleted | Saturday, July 24, 2010 at 07:33 AM

Peter,

>>First, I have no clue what are 'OL-sets' (i.e., ordinary language sets).

This is Bill's term for them - we are halfway through a series of posts about these. In their most recent incarnation here, try this

http://maverickphilosopher.typepad.com/maverick_philosopher/2010/07/sets-pluralities-and-the-axiom-of-pair.html

Briefly, an OL set is what a plural term like 'Those two over there' or 'Alice and Bob' refers to. Everyone seems to agree that 'Alice and 'Bob' refers to just Alice and Bob, and not a third Alice-and-Bob entity different from either of them.

An OL set can also be signified by plural adjectives like 'pair' or 'dozen'. There seems to be agreement, even among the staunch realists here, that 'a pair of shoes' is just a different way of referring to the two shoes.

>>[...] you maintain that a pair is one pair but is not one thing. That seems to me obfuscating. Why isn't a pair a thing? Does a pair of shoes exists? If so, why isn't it a 'thing' just like one shoe is a thing.

A pair can't be a single thing because it is two things, and two things are not a single thing. If I say 'Only a dozen things exist', then I am not saying that 13 things exist (the dozen things plus the dozen).

>>After all a shoe is just a collection of a number of things put together (a sole, and so forth).

Interesting argument. But the shoe is something over and above its physical parts. A dozen things are simply that: a dozen things.

>>Third, as for the ZF-inf matter.

I was replying to your point about the largest number.

Posted by: William | Saturday, July 24, 2010 at 09:49 AM

Richard writes, >>Are we to take it as either self-evident or apodictically proven that nature is finite?<<

Not as far as I can see. How might nature be infinite?

1. Nature (the material universe)would be

potentiallyinfinite if matter were infinitely divisible, i.e., if there were no 'atoms' in the old sense of indivisible bits of matter.2. Nature would be

actuallyinfinite if matter were not only infinitely divisible but actually infinitely divided into as many bits are there are natural numbers, i.e. aleph-nought (aleph-zero, the first transfinite cardinal).3. Nature would be actually infinite if matter were actually divided into as many bits as there are real numbers, i.e. 2 raised to the power aleth-nought.

4. If Big Bang comology is true, then the physical universe is metrically finite in the past direction, meaning that it is a finite number of years (days, seconds, etc.) old. But here is an interesting wrinkle: if time is a continuum, then there is an actual infinity of instants between now and the instant of the Big Bang some 15 billion years ago.

So even on Big Bang cosmology, if space and time are continua, infinity comes into the picture. It seems clear that there cannot be matter without space-time, but presumably there can be space-time without matter.

There are many difficult questions here, especially when we ask how mathematics (which is developed a priori) applies to the empirically given material world.

Posted by: Bill Vallicella | Saturday, July 24, 2010 at 12:54 PM

Peter,

Discussions with William always have a somewhat 'woolly' -- a nice UK expression -- quality to them -- which I blame on him [grin]. But if I understand what the bones of contention are, they are as follows.

I maintain that there are mathematical sets, while Wm. denies this. There is a distinction in the literature between math. sets and commonsense sets (OL sets). It is a distinction which one must make to think clearly about this topic. A housewife might say of some tea cups that they form a set. But that doesn't commit her to the existence of a math. set consisting of the tea cups to which she is referring using the plural expression 'those tea cups.' Here use of 'set' is 'ordinary language.' Same with 'chess set.'

Suppose there are four cups, a, b, c, d. Then I say there are at least five items all having 'ontological status': a, b, c, d, {a, b, c, d}. The fifth item is in addition to the other four. Wm. is aying that in this example there are exactly four 'things' to use his word, a word he uses somewhat idiosynctatically to refer to concrete particulars. The man is a nominalist in the sense of one who maintains that every entity is a concrete particular (not necessarily material). By contrast, I tend to think we need to introduce some 'abstracta' into our ontology. Math. sets are abstracta.

I distinguish between a math. set and its extension. That falls out from there being five items in the tea cup example. As I understand Wm., his thesis, couched in my terminology, is that a math. set reduces to its extension. But here, as elsewhere, reduction boils down to elimination. So his thesis amounts to a denial of math. sets.

But there is another issue on the table, namely, whether, given that there are math. sets, there are also infinite math. sets. What is the argument for positing the latter? Do we need an argument? Can one consistently maintain (i.e., maintain with logical consistency) that there are finite math. sets but no infinite math. sets? Note that an infinite math. set must be an actually infinite math. set.

Posted by: Bill Vallicella | Saturday, July 24, 2010 at 01:22 PM

>>Discussions with William always have a somewhat 'woolly' -- a nice UK expression

'Woolly' means vague and imprecise which I categorically deny. Any fluffy parts around the edges I blame on the spectacles you gentlemen are looking through.

Humph.

Posted by: William | Saturday, July 24, 2010 at 02:04 PM

I mean I would never use an expression like 'ontological status'.

Posted by: William | Saturday, July 24, 2010 at 02:05 PM

I'm sorry if you took offense, but it seems to me that you do not present your theses and arguments in a really clear way.

What is wrong with 'ontological status'? Note that when I used it above I enclosed it in inverted commas. I couldn't say that there are five things: a, b, c, d, {a, b, c, d}, because you use 'thing' in a specific way, to refer to concrete particulars. (Am I right?) On other occasions I have used 'item.'

If you object that 'ontological status' is high-falutin', I won't argue with you. But it is not vague: X has ontological status iff x has being independently of mind and language. That seems clear enough.

Do you agree on what we are disagreeing about? In my tea cup example, I am claiming that there are at least five beings, while you are saying that there are exactly four.

Posted by: Bill Vallicella | Saturday, July 24, 2010 at 02:19 PM

Peter said, >>As to the question about the existence of infinite sets in mathematics (e.g., the set of all natural numbers). The question here is not about the existence of sets as such, but about the existence of sets with infinite cardinality. And while the existence of such sets is stipulated as an axiom and cannot be proven from other axioms, the more fundamental question is what are the consequences and the cost of denying this axiom. For instance, if one denies the existence of an infinite set of natural numbers, but accepts the existence of finite sets, then I challenge such a person to answer the following question: what is the largest member of the set of all natural numbers?<<

"Sets of infinite cardinality" is not a felicitous expression. If the natural numbers form a math. set, then they form an infinite set whose cardinality is not infinite but aleph-nought which is a

definitetransfinite cardinal distinct from other transfinite cardinals.Because there are denumerably infinite and nondenumerably infinite sets, one should not speak of infinite sets as having an infinite cardinality. Similarly, the expression 'infinitely many' is inexact.

Now suppose that there are only finite math. sets. Peter issues the challenge: "what is the largest member of the set of all natural numbers?" Well, if the nat'ls form a set, then they form an infinite set. But if there are no infinite sets, then they do not form a set. So Peter's question rests on a false presupposition.

Of course, there is no greatest nat'l number. All agree on that. So someone who denies that there are infinite sets can say something like this: for any n that you have counted up to, you can alway add 1, and the result will be a nat'l number. We will all agree that the natural numbers are closed under addition. But as far as I can see, it does not straightaway follow that there is a math. set of natural numbers.

Posted by: Bill Vallicella | Saturday, July 24, 2010 at 06:21 PM

>> I'm sorry if you took offense, but it seems to me that you do not present your theses and arguments in a really clear way.

I have said many times before that if you find a thesis or a term unclear, ask me to define it. Implying that 'something somwhere is unclear' is not very helpful. Sometimes, I admit, I give enthymematic arguments omitting certain key assumptions which I take to be obvious. That is different.

I concede there is a weakness in my use of the term 'thing'. I don't want my argument to depend on defining it as 'concrete particular'. A mathematical set is not a concrete particular but it is a thing, if such a thing exists.

My argument is different, and depends on a perfectly normal use of the word 'thing'. What tempts us to say a pair or a dozen are a 'thing' is the singular article 'a', or the quantifier 'one'. 'One pair' of apples must be different from each 'one apple', so there must be a third thing which is one. I reply that it is simply this feature of grammar -the word 'one' in 'one pair' or 'one dozen' - which is misleading. If I assert that only one dozen things exist, I mean that only twelve things exist. It logically follows that if only twelve things exist, namely all of the dozen individual components of the dozen, there can't be one more thing, even though what is designated by 'one dozen' has the number 'one' attached to it.

So my point does not depend on the meaning of the word 'thing' (e.g. whether it covers items or ontological whatsits or whatever). It depends on how we use number words like 'one' with words like 'pair' or 'dozen'. Or 'hundred'. If I say that there are only 'one hundred' things, I clearly do not mean that there are 101 things.

Does that clarify my point, and is my thesis now less fluffy? To reply to your point above, my objection to {a,b,c,d} being a fifth thing does not rest on a peculiar definition of the word 'thing'. It would be a thing, if there were such a thing. But I propose that there isn't.

On a linguistic note 'flannel' is an older British english expression which is now obselete, replaced by the crude US english 'bullshit'. As in 'he is flanneling again'. Can be a noun also, as with its US counterpart.

Posted by: William | Sunday, July 25, 2010 at 12:40 AM

Just a quick note on Bill's comments:

"So someone who denies that there are infinite sets can say something like this: for any n that you have counted up to, you can alway add 1, and the result will be a nat'l number."

Not quite. Someone who is inclined to say that will have to answer three questions:

(a) who is the 'you' here?

(b) what is the 'can' here?

(c) what is the domain of 'always' here?

After all (i) if the 'you' is a mortal being such as us human beings, or any other temporary entity; and (ii) the 'can' means whatever actions are available within a finite lifespan; and (iii) 'always' means at any time during a finite lifespan; then of course you will get a last number for obvious reasons.

Posted by: Account Deleted | Sunday, July 25, 2010 at 10:21 AM

Peter,

I didn't use the phrase 'last number.' I said that all agree that there is no greatest natural number. The question is whether this obvious truth must be interpreted so as to entail that there is an actually infinite set of natural numbers. I say No. It need not be interpreted so as to have that entailment. 'No greatest natural number' can be interpreted to mean that a computer (human or otherwise), starting from 0 and adding 1 will generate a series 0, 1, 2, 3 . . . n, n+ 1, . . . such that if the computing process were to go on indefinitely no upper bound would ever be reached.

Of course, at any given time t there will be some n such that n is the result of the computation at t, e.g. 568,972. But this LAST natural number is not the GREATEST natural number.

I think what I am saying is equivalent to saying that the nat'l number series is infinite, but only potentially infinite. This is consistent with the trivial truth that there is no greatest natural number. And because it is consistent with it, you cannot validly infer that that there is a set of naturals from the trivial truth.

Posted by: Bill Vallicella | Sunday, July 25, 2010 at 01:10 PM

William,

Your thesis is clear. A dozen eggs is 12 things, not 13 things. I was wrong about your use of 'thing.' You mean it as interchangeable with 'entity.' OK. Good.

Suppose there are 12 eggs on the table. It would be silly were one to say, 'I see the 12 eggs, but where is the dozen?' Obviously, if there is a math set consiting of the 12 eggs it is not wholly distinct from the 12 eggs inasmuch as it has them as its members. It cannot exist without them. The relation of set to members is not like the relation of carton to eggs: the carton can exist without the eggs. I am sure you will agree with this.

The dispute is whether in reality (apart from our use of expressions like 'a dozen' or 'one dozen' and apart from any acts of mental collection) there is a 'whole' that has the 12 eggs as 'parts,' a whole that is distinct -- but not WHOLLY distinct, see preceding para. -- from the 12 eggs.

Do you agree that that is the gravamen of the dispute?

Posted by: Bill Vallicella | Sunday, July 25, 2010 at 01:32 PM

Bill,

I do not think that "There is no greatest NN" can be so interpreted without presupposing somewhere the existence of some actual infinite series. For instance, the computer example will not work because every computer will in fact break down and stop working at some point or because the physical universe will turn into dust, etc. So you need to introduce some sort of counterfactual such as the one you suggested: "...if the computing process were to go on indefinitely no upper bound would ever be reached." But, now, what is meant here by "indefinitely"? You cannot explicate this notion in terms of an actual time series because an actual time series is infinite.

I understand that you are attempting to use the notion of potential infinity in order to show that the truth that there is no greatest NN is compatible with the denial that there exists an actual infinite set. But I think that the notion of potential infinity is more problematical than positing the existence of an infinite series because it relies on the idea that something that is inherently finite (anything in the physical universe) has the potential to go on essentially forever.

While I think that the notion of potential is ultimately indispensable when it comes to the physical realm, it cannot carry the weight that is required by the notion of a potential infinity as a substitute for the existence of an infinite set.

Posted by: Account Deleted | Sunday, July 25, 2010 at 02:16 PM

I have been banging my head against a paper (Kaye & Wong "On interpretations of arithmetic and set theory") linked by William back in one of his comments on this thread in which it is proven that PA (Peano arithmetic) and ZF together with the negation of the axiom of infinity are mutually interpretable. I thank William for the link.

The technical aspects of the paper are well beyond my skills in set theory and model theory. However, I find the conclusion and the general "folklore result" very puzzling. It sort of reminds me of a similar puzzling result known as Skolem's Paradox: i.e., that if a theory which says that there are uncountable sets has a model at all, then it has a countable (first order) model. How can a theory which says that there are uncountable sets be true in a countable model? What happened to the uncountable sets in such a model?

Well, due to the apparent similarity I have discerned in the case of Skolem's Paradox and the Folklore results regarding ZF-inf, I proceeded to read a paper in Stanford Encyclopedia by Timothy Bays called "Skolem's Paradox". I recommend it to all. It is very clear and very clearly explains the pitfalls that await us if we take model theoretic results at face value without critical scrutiny. It is quite unbelievable how misleading model theoretic results could be if not approached with caution. I do not know whether the paper by Kaye & Wong are subject to similar concerns of interpretation, but they certainly may be. And other issues may be lurking in the background. Of course, I do not mean to challenge the technical results, only their philosophical significance.

Posted by: Account Deleted | Sunday, July 25, 2010 at 05:13 PM

Peter writes >>But I think that the notion of potential infinity is more problematical than positing the existence of an infinite series because it relies on the idea that something that is inherently finite (anything in the physical universe) has the potential to go on essentially forever.<<

But potential infinity does not requite that any physical thing have the potential or power to go on doing something forever.

The idea is simply this: there is no greatest nat'l number. Why do you think that that entails that there is a set of natural numbers? What's the argument?

Posted by: Bill Vallicella | Sunday, July 25, 2010 at 07:34 PM

Bill,

I am currently not saying that the proposition that there is no greatest NN *entails* that there is an infinite set of NN. What I am trying to argue instead is that I do not see how it is possible to construe this fact in terms of potential infinity. However, it is perhaps possible to get an equivalent notion in terms of hereditary finite set theory and perhaps this is what you are getting at. But I am not familiar with this technical notion, so I cannot comment on it as yet.

Posted by: Account Deleted | Sunday, July 25, 2010 at 09:04 PM

>>The dispute is whether in reality (apart from our use of expressions like 'a dozen' or 'one dozen' and apart from any acts of mental collection) there is a 'whole' that has the 12 eggs as 'parts,' a whole that is distinct -- but not WHOLLY distinct, see preceding para. -- from the 12 eggs.

Do you agree that that is the gravamen of the dispute?

I confess I had to look up 'gravamen'. Yes, it is the crux of the dispute.

Posted by: William | Monday, July 26, 2010 at 12:47 AM

>> I proceeded to read a paper in Stanford Encyclopedia by Timothy Bays called "Skolem's Paradox". I recommend it to all.

Now that would be an interesting one to discuss. I had a correspondence with Bays some years ago, about the same time Tom and I were talking about pluralities.

Posted by: William | Monday, July 26, 2010 at 12:48 AM

William,

And what was the topic of the correspondence with Bays? Skolem's Paradox, pluralities, or something else? I downloaded a bunch of his papers on Skolem etc. Planning to read the stuff as time permits.

Posted by: Account Deleted | Monday, July 26, 2010 at 04:52 AM

William,

Here is a question for you. Sets are indispensable in order to do model theory. I cannot imagine how one would do model theory without sets. Since you deny the existence of sets, are you also willing to discard model theory? Or do you think that model theory can be done without sets? If so, then I would like to know in a general way how such model theory would work.

Posted by: Account Deleted | Monday, July 26, 2010 at 10:54 AM