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.
Leave a Reply to Justin Davis Cancel reply