Sets, Pluralities, and the Axiom of Pair

In a thread from the old blog, resident nominalist gadfly 'Ockham'/'William' made the fascinating double-barreled claim that:

. . . (a) there are such things as sets and (b) the axiom of pairs is false. Briefly, I claim that 'a set of x's' is just another way of saying 'those x's'. The fundamental error of set theory is using a logically singular expression {a, b} to refer to what in ordinary language a plural term refers to, using an expression such as 'a and b' or similar.

I take O to be saying that there are sets, but they are not the sets we read about in standard treatments of axiomatic set theory, and whose properties are all and only the properties ascribed to them in axiomatic set theory, Zermelo-Fraenkel with Choice, to be specific. Suppose we call the latter mathematical sets, and the former ordinary language (commonsense) sets. Then what O is claiming is that there are ordinary language (OL) sets, but there are no mathematical sets. That there are no mathematical sets on O's view follows from O's denial of the Axiom of Pair, a crucial ingredient of ZFC. Here is a formulation of the latter:

PAIR. Given any x and y, there is a set {x, y} the members of which are exactly x and y.

X and y can be either sets or nonsets. So given that Socrates exists and that Plato exists, it follows by PAIR that a third item exists, namely, {Socrates, Plato}. (I use 'there is' and 'there exists' interchangeably.) That a third item exists is what I affirm and what O denies. For O, the plural term 'Socrates and Plato' does not refer to a single third item, the set consisting of Socrates and Plato; and yet it does refer to something, a thing that is an ordinary language set. For O, there are exactly two items in our example, Socrates and Plato, and not three, as I claim.

Let us say that the referent of a plural term such as 'Socrates and Plato' or 'the British Empiricists' or 'the Hatfields' is a plurality. A plurality is an ordinary language set. A gaggle of geese, a pride of lions, a coven of witches, a bunch of grapes, a pack of wolves -- these are all pluralities or OL sets. That there are OL sets, or pluralities, is presumably not in dispute. Nor, I think, could anyone rationally dispute their existence. That there is such a thing as a pair of shows cannot be reasonably denied; that the two shoes form a mathematical set can be reasonably denied at least prima facie.

If I understand O, he is saying that all reference to sets is via plural referring expressions such as 'these books,' 'Dick Dale and the Deltones,' 'the barristers of London,' etc. There is no reference to any set via a singular referring device such as the singular definite description, 'the set consisting of these books.'

Now consider the question whether there are sets of sets. I claim that it is a fact that there are sets of sets, and that this fact causes trouble for O's nominalist view that all sets are pluralities. Consider the Hatfields and the McCoys. These are two famous feuding Appalachian families, and therefore two pluralities or OL sets. But there is also the two-membered plurality of these pluralities to which we refer with the phrase 'the Hatfields and the McCoys' in a sentence like 'The Hatfields and the McCoys are feuding families.'

If, however, a plurality of pluralities has exactly two members, as in the case of the Hatfields and the McCoys, then the latter cannot themselves be pluralities, but must be single items, albeit single items that have members. That is to say: In the sentence, 'The Hatfields and the McCoys are two famous feuding Appalachian families,' 'the Hatfields' and 'the McCoys' must each be taken to be referring to a single item, a family, and not to a plurality of persons. For if each is taken to refer to a plurality of items, then the plurality of pluralities could not have exactly two members but would many more than two members, as many members as there are Hatfields and MCoys all together. Compare the following two sentences:

1. The Hatfields and the McCoys number 100 in toto.

2. The Hatfields and the McCoys are two famous feuding Appalachian families.

In (1),'the Hatfields and the McCoys' can be interpreted as referring to a plurality of persons as opposed to a mathematical set of persons. But in (2), 'the Hatfields and the McCoys' cannot be taken to be referring to a plurality of pluralities; it must be taken to be referring to a plurality of two single items.

Or consider the following said to someone who mistakenly thinks that the Hatfields and the McCoys are one and the same family under two names:

3. The Hatfields and the McCoys are two, not one.

Clearly, in (3) 'the Hatfields and the McCoys' refers to a two-membered plurality of single items, each of which has many members, and not to a plurality of pluralities. And so we must introduce mathematical sets into our ontology.

This is connected with the fact that '___ is an element of . . .' in axiomatic set theory does not pick out a transitive relation: If x is an element of y, and y is an element of z, it does not follow that x is an element of z. Socrates, a nonset, is an element of various sets; but he is clearly not a member of any of these set's power sets. (The power set P(S) is the set of all of S's subsets. Clearly, no nonset can be a member of any power set.) But if there are no mathematical sets, and every set is a plurality, then it seems that the elementhood or membership relation would be transitive. A set of sets would be a plurality of pluralities such that if x is an element of S and S an element of S *, then x is an element of S*. My conclusion, contra 'Ockham,' is that we cannot scrape by on OL sets, or pluralities, alone. We need mathematical sets or something like them: entities that are both one and many.

REFERENCES

Max Black, "The Elusiveness of Sets," Review of Metaphysics, vol. XXIV, no. 4 (June 1971), 614-636.

Stephen Pollard, Philosophical Introduction to Set Theory, University of Notre Dame Press, 1990.

Comments

[Jan]

Excellent post. Thanks.

[Bill Vallicella]

Thanks, Jan. I hope you are well. I take it you thought the above argument for mathematical sets is sound. Perhaps you can help me respond to my nominalist friend when and if he surfaces.

[Justin Davis]

I've recently stumbled upon your site and realized I have precious little familiarity with analytic philosophy. Upon reflection this makes sense because I am an undergraduate in a post-modern dominated school. I guess I simply didn't realize how ignorant I was. My question is, where does one start if they want to do a thorough, ongoing study of analytic philosophy? An acquaintance, who is the only analytic philosopher at the school, suggested I just start with Wittgenstein, but that seems a little flippant.

[Bill Vallicella]

Justin,

You should start with Frege, and then proceed to Russell and Moore.

[William]

Justin - seconding what Bill says, but particularly recommending The Frege Reader by Michael Beaney. Has good discussion around the work of Frege, plus many relevant extracts from Frege's writing. Russell's work is voluminous, I recommend secondary literature on his Theory of Descriptions. For example Meaning and Reference edited by A.W.Moore, which contains all the classic texts of Analytic Philosophy.

I don't recommend Wittgenstein for a beginner.

[William]

Bill (sorry not to have surfaced earlier, Friday is family-time).

Rather than a cudgel, or a Maxim gun, I brought my little razor. We need to add suitable 'units' to a plurality. Thus 'a' pair of things is two things, but one pair of things. In reply to your point that 'the Hatfields and the McCoys' cannot be taken to be referring to a plurality of pluralities; it must be taken to be referring to a plurality of two single items, I reject that. One family is not one thing, but a number of things (people).

You give the example "The Hatfields and the McCoys are two, not one." Again, units required. They are two families, not one. They are 100 people. In summary, the word 'thing' picks out our most elementary basic unit: the referent of a singular term. Plural terms pick out several things, and never one thing. Id quod unum est, non est plura. But it can refer to one F, where 'F' picks out a suitable unit ('pair', 'dozen', 'family', 'God').

On your argument that mathematical set-membership is not a transitive relation. Correct. But that is a definition, not an argument for the existence of mathematical sets. I have argued that if things exist, OL sets exist, as a logical consequence. But the existence of M sets is not a logical consequence. We need existence axioms such as the Pair Axiom.

[Justin Davis]

Bill, William,

Thanks to you both. I appreciate it.

[Bill Vallicella]

Justin,

Agreeing with William, I would add that you won't understand much of W's Tractatus without having read Frege and Russell, which is why you can't start with W. As for the Phil. Inv., they make sense only as a rejection of the sort of approach represented in the Tractatus.

But to fully contextualize analytic phil and its rise, one has to know something about Bradley and Co. against whom Russell and Moore rebel.

You also have to know something about trad logic in order to understand Frege's logical innovations.

[William]

You won't understand much of W's Tractatus even if you have read Frege and Russell.

[William]

Meanwhile, back to sets. Actually there is non-transitivity even with OL sets. Suppose a is one of the pair a-b. And a-b is one pair of the two pairs a-b, c-d. Then it does not follow that a is one pair of a-b, c-d. It is one thing among them, but not one pair among them.

What OL set theory would look like I don't know. I haven't decided whether an infinite OL set could exist or not. For example, in M set theory it is possible to construct a domain where there are infinitely many M-sets, but no set containing all of them. You simply postulate

(*) For every Mset S, there is at least one element e such that e notin S

Assuming there is at least one set, that guarantees at least one element not in it. This combines to form a further set, and thus a further element not in it, and so on ad infinitum. For the same reason, there cannot be an Mset of all the elements in the domain. For there would have to be at least one element not in it, and so it would not be the set of all elements.

However, Mset theory allows us to add a set S* of all the elements by modifying (*) as follows

(*) For every S in S*, there is at least one element e in S* such that e notin S

This is the classical axiom of infinity, given as the 7th axiom in Zermelo's original theory http://en.wikipedia.org/wiki/Zermelo_set_theory. Thus in M set theory, we can suppose that an infinite domain D does or does not contain a set of all the elements of D. Whether there is such a set is contingent.

I suspect we can't do the same in OL set theory, because it is not similarly contingent, but rather a matter of logic as to whether a set of elements exist. For an OL set simply is is its elements. Deny the existence of an OL set, and you deny the existence of at least one of its elements. Create an OL set, and you must create at least one element that is 'one of' it.

This obviously leads to a problem. If the existence of an OL set corresponding to a number of elements is not contingent, this means that the axiom of infinity in OL set theory is not contingent either. If it is false, it is necessarily false, in which case OL set theory has no infinite set (even if an infinite number of elements exist), and so no set hierarchy. If it is true, it is necessarily true. But suggests further problems I haven't really explored. If there is a hierarchy in OL set theory, and if there is always an OL set corresponding to any elements of the hierarchy, then there must be a set of all elements of the hierarchy. But that leads to paradox, as we know http://en.wikipedia.org/wiki/Burali-Forti_paradox .

But it's some time since I looked at this (and apologies if I have made any mistakes here - it is some time since I looked at classical set theory).

[David Brightly]

I think that the discussion so far tends to show that there is a degree of ambiguity in natural language plural terms. Sometimes they seem to refer to the elements making them up and sometimes to the plurality as a whole. This comes out in the verb number that they take. Consider: 'The orchestra plays better under their new conductor'. This solecism (if such it is) usually passes without comment in ordinary speech. Or: 'My family is large' versus 'My family are large'. The discussion also suggests that natural language has few resources for expressing pluralities of pluralities. The best we can do is to use the generic 'pair' (two of a single kind) in conjunction with a plural term as in Bill's 'pair of feuding families'. Again, there is room for debate over exactly what such a construction refers to.

In light of this I'd like to suggest that we should see the apparatus of mathematical sets as an artificial device for refining and extending the plural referencing capabilities of natural language. We could call them 'polyrefs'. Polyrefs form a refinement because the singular/plural ambiguity is eliminated: a polyref is strictly singular. Suppose the polyref S refers to 2 and 4. We write this as S = {2, 4}. We do not say 'S are even'. Nor do we say 'S is even', since S, a polyref, is not the kind of thing that can be even. Informally, we might say 'The Ss are even'. More formally, to refer to the referents of S we use the apparatus of quantification: for all x in S, x is even. Polyrefs form an extension because polyrefs of polyrefs of polyrefs ... to any required hierarchical level are readily constructed and have precise meaning.

If we see polyrefs as a linguistic referencing mechanism then the question of their existence falls away. Since they are artificial there remains the engineering question, Do they work reliably? In this context, Do they lead us into contradiction? The applied mathematics of polyrefs, aka Axiomatic Set Theory tries, in part, to answer this question. Lastly, in response to WW's claim regarding the pairing axiom, I think we should say this: The pairing axiom is not making a conventional existence assertion. What it is saying is that if referents a and b exist or are themselves polyrefs, then it's safe to use the polyref {a, b}. Safe in the sense of not leading to contradiction. The strange status of the Axiom of Choice now makes more sense. Unlike other uses of the axiomatic method in mathematics, the system of ZF is not trying to capture truths about some class of entity. Rather, it's an engineering specification that we have drawn up. We know that AC is consistent with the other axioms of ZF, so we are free to use a referencing mechanism that satisfies AC or one which does not, according to choice (!)

[Bill Vallicella]

W,

I don't see that you have met my argument. I admit that my presentation was not very clear. Compare

1. The British Empiricists and the Continental Rationalists are 6 in number. (Locke, Berkeley, Hume, Descartes, Spinoza, Leibniz)
2. The British Empiricists and the Continental Rationalists are 2 in number.

Both sentences are true. The same plural expression-type occurs in subject position in both sentences: 'The British Empiricists and the Continental Rationalists.' But in its first occurrence it refers to the six gentlemen mentioned, while in its second occurrence it refers to something that has mathematical sets as members.

[William]

>>2. The British Empiricists and the Continental Rationalists are 2 in number.

I don't see the 'suitable unit' here. Two of what? They are 6 of people. Perhaps you mean '2 schools'. But then my argument kicks in. 2 schools of philosophers can be 6 philosophers, but not 2 philosophers.

>>the system of ZF is not trying to capture truths about some class of entity. Rather, it's an engineering specification that we have drawn up.

Very true, and leads (in my experience) to frustrating arguments between set theorists and philosophers. Philosophers in general are trying to capture truths about classes of entity.

[William]

>>Sometimes they seem to refer to the elements making them up and sometimes to the plurality as a whole.

I have already argued: both. 'this one pair of things' refers to the same things as 'these two things'. We get linguistically confused because of the 'one' in 'one pair', and so we imagine that it also means 'one thing'. One pair is not one thing. It is two things.