Friday, September 4, 2009

C-sets and L-sets. Draft

C-sets. or Cantorian sets, are the usual sets of set theory. They were first (for practical purposes) described by Cantor in his work with infinity, which gave them a slightly shady reputation, enhanced by the discovery of an array of paradoxes in the earliest attempts to axiomatize the theory.  Eventually,number of axiomatizations that avoided the known paradoxes in various ways were devised.  All the various axiomatizations are incomplete, of course, since arithmetic is and can be defined within them. They also differ in various ways at their outer edges (is the next largest cardinal after Aleph-null, that of the set of natural numbers, R, that of the set of reals?). But they agree in the basic area we are concerned with, namely, simple set, their subsets, and simple operations on these. Briefly, a set is different from its members, in particular, {a} (the set whose only member is a) is different from a (the thing in it). Further, from a given set, {abc}, say. we can get other sets using only the members of the original set (its subsets): in this case {a}, {b}, {c}, {ab}, {ac}, {bc}, and, for completeness, the original set {abc} and the empty set{}, which has no members. The order in which the members are listed is not significant, but the fact that these members are in different sets will be: {{a}{b}} is different from {ab}, of course, and {{ab}{ac}} is different from {abc}, and so on. A set, once constructed, can now behave as an element in further sets, as just applied, and so, all of the subsets can be gathered together into a single set (called the power set because its size for a set with n members is 2 to the power n). In this way, quite large sets can be built from relatively small beginnings (the natural numbers, in one approach, are built up from the empty set -- about as small as you can get -- by taking its power set, {{}}, joining the two into a new set, {{}{{}}}, then combining this set with its members to form a set with three elements, and so on forever).

L-sets were developed first by Lesniewski and independently by Leonard and Goodman (and Quine).. They are mathematically more untidy that C-sets (there is no empty set, so a set with n members has only 2^n -1 subsets) and also less useful mathematically (it is hard to get bigger sets). But they prove to be more useful in representing many situations in language (and thus expanding logic a bit). For L-sets, {a} is the same as a, further, {{a}{ab}} collapses to {ab}, {{ab}{c}} = {{a}{bc}} = {{a}{b}{c}} = {abc}, and so on. That is, a set is always a set of its ultimate components, even if we talk about intermediate subsets.

In set theories, of course, sets are individuals (the values for variables, the referents of constants, and so on) and so, they have properties and enter relations. But, in the theory, these attributes tend to be either dummies or set-theoretic ones. Little is said about sets and ordinary properties and relations , "carries a piano", say, or "wins a race." But, if we have any intuitions about sets, they don't seem to be the sorts of thing that carry anything or even enter races, even though their members might be.

And yet, something formally very like sets do these thing in everyday language: "The boys carried the piano" need not mean that each carried it by himself; it might mean that they carried it together, acting formally as a set. (Each of the boys participated in carrying the piano, though each's exact role is unspecified.) And, of course, a team (looks like a set in a general way -- a number of things conceived of as together) can win a race even if only one member runs (the other participate by being on the same team, I suppose -- that is a characteristic of teams among sets). And it turns out in many cases to simplify thing quite a bit to take plural references as to sets, rather than somehow to each of the several individuals we take as making up the set. We can disambiguate if we need to, but often it does not matter (as the race case suggests). So long as the piano gets carried, we don't care how the boys do it. If we really want to specify that each carried it alone part of the way, then we can say so:'"each of the boys carried the piano."

While we could do this sort of thing with either C-sets or L-sets, L-sets seem the more natural. Bunches of things such as we have in mind don't seem to grow into bigger bunches by subdividing and recombining, Further, if we are mistaken about the referent being plural, the same pattern applies since {a}=a, the singleton set reduces to its member. Even the lack of an empty set, so damaging to the mathematical uses, proves useful here, since a reference to a something having a property which nothing in fact has automatically renders a sentence false for L-sets, but makes a reference to the null-set in C-sets and the null set has many properties, mostly irrelevant to whatever we were talking about or relevant but holding only through logical tricks -- neither desirable situations.

It should be noted that some people raise objections to using L-sets to treat languages. The objection runs that doing so compels the language to implicitly recognize the existence of such things as L-sets, since they are necessarily in the range of quantifiers in the language. While I am not sure why this is a problem, especially in a language, like English, which regularly interchanges L-set words like "bunch" and "group" with simple plurals, I give way to others' quest for ontological purity and say, that we ought not think of L-sets as something different (or over and above) the members considered together. This is, of course, very easy to do, given the transparency of L-sets. It requires some changes in the way we describe the logic of the situation, but only minor ones. In particular, we need to allow that quantifiers may take a number of instances simultaneously and together and that we have a way of pulling out particular instances from these. That being done, we can deal with plurals as really being about several things rather than one thing of which the several are members, which does seem more natural. The point is that the logics of the two approaches are exactly the same and even the two metalanguages, while not the same, are directly translatable the one into the other and so totally congruent. As an old-timer raised on set theory, I stick to the locution I am most comfortable with, but I try always to include the other reading as well.

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