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.

A Cantorian Argument Why Possible Worlds Cannot be Maximally Consistent Sets of Propositions

A commenter in the 'Nothing' thread spoke of possible worlds as sets.  What follows is a reposting from 1 March 2009 which opposes that notion.

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CANTOR_OCT20_G_290w_q30 In a recent comment, Peter Lupu bids us construe possible worlds as maximally consistent sets of propositions.  If this is right, then the actual world, which is of course one of the possible worlds,  is the maximally consistent set of true propositions.  But Cantor's Theorem implies that there cannot be a set of all true propositions. Therefore, Cantor's theorem implies that possible worlds cannot be maximally consistent sets of propositions.

1. Cantor's Theorem states that for any set S, the cardinality of the power set P(S) of S > the cardinality of S. The power set of a set S is the set whose elements (members) are all of S's subsets. Recall the difference between a member and a subset. The set {Socrates, Plato} has exactly two elements, neither of which is a set. Since neither is a set, neither is a subset of this or any set. {Socrates, Plato} has four subsets: the set itself, the null set, {Socrates}, {Plato}. Note that none of the four sets just listed are elements of {Socrates, Plato}. The power set of {Socrates, Plato}, then, is {{Socrates, Plato}, { }, {Socrates}, {Plato}}.

On the Abstractness of Mathematical Sets

Let us agree that x is concrete iff x is causally/active passive and abstract otherwise.  Many say that mathematical sets ('sets' hereafter: 'mathematical' as opposed to 'commonsense') are abstract objects, abstract entities, abstracta.  Why?

Argument One:  In set theory there are singleton sets, e.g. {Quine}.  Obviously, Quine is not identical to {Quine}.  The second is a set, the first is not.  Yet the difference cannot be the difference between two concreta.  Quine is a concretum.  Therefore, {Quine} is an abstractum.  This is of course meant generally: singletons are abstracta.  Now if singletons are abstracta, then all sets are. 

Argument Two:  In set theory there is a null set.  It is not nothing, but something despite having no members. Yet it cannot be a concrete something.  Therefore, it is an abstract something.  And if one set is abstract, all are.

Contra Argument One:  A statue and the lump of clay that constitute it are numerically distinct.  (For the one has properties the other doesn't have, e.g., the lump, but not the statue, can exist without having the form of a statue.)  And yet both are concrete, i.e., both are causally active/passive.  If this is possible, why should it not also be possible that Quine and {Quine} both be concrete?  One could say that Quine and {Quine} occupy the same 'plime' to borrow a term form D. C. Williams, the same place-time, in the way statue and lump do.

Contra Argument Two:  Possibly, there is a concrete atomic entity. Being atomic, it has no parts.  So why should a set's having no members rule out its being concrete?

Are any of these arguments compelling?

Another Example of a Necessary Being Depending for its Existence on a Necessary Being

The Father and the Son are both necessary beings.  And yet the Father 'begets' the Son.  Part, though not the whole, of the notion of begetting here must be this: if x begets y, then y depends for its existence on x.  If that were not part of the meaning of 'begets'' in this context, I would have no idea what it means.  But how can a necessary being depend for its existence on a necessary being?  I gave a non-Trinitarian example yesterday, but it was still a theological example. Now I present a non-theological example.

I assume that there are mathematical (as opposed to commonsense) sets.  And I assume that numbers are necessary beings.  (There are powerful arguments for both assumptions.) Now consider the set S = {1, 3, 5} or any set, finite or infinite, the members of which are all of them necessary beings, whether numbers, propositions, whatever.  Both S and its membership are necessary beings.  If you are worried about the difference between members and membership, we can avoid that wrinkle by considering the singleton set T = {1}.

T and its sole member are both necessary beings.  And yet it seems obvious to me that one depends on the other for its existence:  the set is existentially dependent on the member, and not vice versa.  The set exists because — though this is not an empirically-causal use of 'because' — the members exist, and not the other way around.   Existential dependence is an asymmetrical relation.  I suppose you either share this intuition or you don't.  In a more general form, the intuition is that collections depend for their existence on the things collected, and not vice versa.  This is particularly obvious if the items collected can also exist uncollected.  Think of Maynard's stamp collection.  The stamps in the collection will continue to exist if Maynard sells them, but then they will no longer form Maynard's collection. The point is less obvious if we consider the set of stamps in Maynard's collection.  That set cannot fail to exist as long as all the stamps exist.  Still, it seems to me that the set exists because the members exist and not vice versa.

And similarly in the case of T.  {1} depends existentially on 1 despite the fact that there is no possible world in which the one exists without the other.  If, per impossibile, 1 were not to exist, then {1} would not exist either. But it strikes me as false to say: If, per impossibile, {1} were not to exist, then 1 would not exist either.  These counterfactuals could be taken to unpack the sense in which the set depends on the member, but not vice versa.

It therefore is reasonable to hold that two necessary beings can be such that one depends for its existence on the other.  And so one cannot object to the notion of the Father 'begetting'  the Son by saying that no necessary being can be existentially dependent upon a necessary being.  Of course, this is not to say that other objections cannot be raised.

Trinity and Set Theory

Let S and T be mathematical sets. Now consider the following two propositions:

1. S is a proper subset of T.

2. S and T have the same number of elements.

Are (1) and (2) consistent? That is, can they both be true? If yes, explain how.

If you think (1) and (2) are consistent, then consider whether there is anything to the following analogy. If there is, explain the analogy. There is a set G. G has three disjoint proper subsets, F, S, H. All four sets agree in cardinality: they have the same number of elements.

A Cantorian Argument Why Possible Worlds Cannot be Maximally Consistent Sets of Propositions

In a recent comment, Peter Lupu bids us construe possible worlds as maximally consistent sets of propositions.  If this is right, then the actual world, which is of course one of the possible worlds,  is the maximally consistent set of true propositions.  But Cantor's Theorem implies that there cannot be a set of all true propositions. Therefore, Cantor's theorem implies that possible worlds cannot be maximally consistent sets of propositions.

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Transitive Sets and the Distinctness of Sets From Their Members

Vlastimil asked for examples of transitive sets.  A transitive set is a set every element of which is a subset of it.  (Hrbacek and Jech, Introduction to Set Theory, p. 50) There is no lack of examples.  The null set vacuously satisfies the condition 'if x is an element of S, then x is a subset of S.'  The set consisting of the null set — {{ }} — is also transitive:  it has exactly one element, the null set, and that element is a subset of it because the null set is a subset of every set.

Now consider the set consisting of the foregoing two sets, the null set and the set consisting of the null set:  {{ }, {{ }}}.  This set has two elements and both are subsets of it.  The null set is a subset of every set, and the set consisting of the null set is also a subset of it in virtue of the fact that the null set is an element of it.

If we identify 0 with the null set, and 1 with the set consisting of the null set, and 2 with the set consisting of the null set and the set consisting of the null set, then 3 will be the set whose elements are the elements of 0, 1, and 2 which is:  {{ }, {{ }}, {{ }, {{ }}}}.  This last set has three elements and each is a subset of it.  One can continue like this and generate as many transitive sets as one likes.  For each natural number there is a corresponding transitive set.

Now how does all this bear upon my assertion that a (mathematical) set is an entity 'over and above' its members (elements)?  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.

Vlastimil seems to think that the existence of transitive sets is somehow at odds with the claim that sets are distinct from their members.  Or perhaps he thinks that some sets are distinct from their members and some are not.  So consider {{ }, {{ }}}.  This is a transitive set since every member of it is a subset of it, which is equivalent to saying that every member of a member of it is a member of it.  Thus { } is a member of {{ }}, which is a member of {{ }, {{ }}}.  But although every member of the set in question is a subset of it, this does not alter the fact that the set is distinct from its members.

So I'm not sure what Vlastimil is driving at.

Note that if every member of a set is a subset of it, this is not to say that every subset of it is a member of it.  {{ }, {{ }}} has itself and {{{ }}} as subsets but not as elements.  Only if there were a set all of whose members are subsets of it and all of whose subsets are members of it could one argue that there are sets for which the membership and subset relations collapse, and with it the distinction between a set and its members.

On the Elusive Notion of a Set: Sets as Products of Collectings

In an important article, Max Black writes:

Beginners are taught that a set having three members is a single thing, wholly constituted by its members but distinct from them. After this, the theological doctrine of the Trinity as "three in one" should be child's play. ("The Elusiveness of Sets," Review of Metaphysics, June 1971, p. 615)

1. A set in the mathematical (as opposed to commonsense) sense is a single item 'over and above' its members. If the six shoes in my closet form a mathematical set, and it is not obvious that they do, then that set is a one-over-many: it is one single item despite its having six distinct members each of which is distinct from the set, and all of which, taken collectively, are distinct from the set.   A set with two or more members is not identical to one of its members, or to each of its members, or to its members taken together, and so the set  is distinct from its members taken together, though not wholly distinct from them: it is after all composed of them and its very identity and existence depends on them.

In the above quotation, Black is suggesting that mathematical sets are contradictory entities: they are both one and many.  A set is one in that it is a single item 'over and above' its members or elements as I have just explained.  It is many in that it is "wholly constituted" by its members. (We leave out of consideration the null set and singleton sets which present problems of their own.)  The sense in which sets are "wholly constituted" by their members can be explained in terms of the Axiom of Extensionality: two sets are numerically the same iff they have the same members and numerically different otherwise. Obviously, nothing can be both one and many at the same time and in the same respect.  So it seems there is a genuine puzzle here.  How remove it?

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Non-Empty Thoughts About the Empty Set

1. The empty or null set is a strange animal. It is a set, but it has no members. This is of course not a contingent fact about it, but one bound up with its very identity: the null set is essentially null. Intuitively, however, one might have thought that a set is a group of two or more things. Indeed, Georg Cantor famously defines a set (Menge) as "any collection into a whole (Zusammenfassung zu einem Ganzen) of definite and separate objects of our intuition and thought." (Contributions to the Founding of the Theory of Transfinite Numbers, Dover 1955, p. 85) In the case of the null set, however, there are no definite objects that it collects. So in what sense is the null set a set? One might ask a similar question about singletons, sets having exactly one member. But I leave this for later.

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