Thursday, June 21, 2007

Cantor's Paradox

Given some things, we have not only those things but also that collection (the plurality as another unity, the many as one), so the question arises, is a collection (so to speak) with just one member also a further thing, over and above its only member? Intuitively it is not, but it is often (as below) tidier mathematically to say that it is, and then (if we also outlaw self-membership, and potentially infinite collections, etc.) then our collections are sets (and their members are elements). Cantor’s paradox concerned his sets.
......Another question is, when does one collection (a whole number of things) have as many members as another? Cantor realised that it was when the members of one collection correlate one-to-one with those of the other, when they have the same cardinal number of members. The most basic numbers are the natural numbers, 1, 2, 3 and so forth (adding 1 ad infinitum), so how many of them are there? The classical answer was that, since they are generated endlessly (are potentially infinite) they have no number, but Cantor’s answer was that there are aleph-null of them, where aleph-null was the first of his transfinite cardinal numbers.
......Similarly there are aleph-null rational numbers (since they correlate one-to-one with the natural numbers) but more real numbers—there are as many endless sequences of 0s and 1s as there are real numbers (which we may think of as endless decimal or binary expansions), and hence as many subsets of the set of natural numbers (since we may think of each 1 as signifying membership of that set, and each 0 non-membership), and hence there are more real numbers, as follows.
......Counting a singleton (a set with a single element) as different to its element tidies up the mathematics of the set of all the subsets (the power-set) of a given set. E.g. if S = {a, b, c} then its power-set, P(S) = {{a, b, c}, {a, b}, {b, c}, {a, c}, {a}, {b}, {c}, {}}. In general, if the size of S is # (e.g. 3) then the size of P(S) is 2 to the power of # (e.g. 8). Cantor showed that the power-set of any set (even one of transfinite size) is bigger than that set by showing that there could not be any bijection between that set and its power-set (via his diagonal argument), so that since, for each element of that set, there is a singleton that is an element of its power-set, hence the power-set is bigger.
......Cantor’s paradox concerned the set of all the sets. Although it would be the biggest set (by its definition), its power-set would be bigger (by the diagonal argument), whence there is no such set. And although that seems no worse than Russell’s paradox, consider the pure sets. They are built out of (by taking collections of (collections of, etc.)) empty sets, where the empty set does not have a single element (e.g. {} above). Within any particular set theory, the unique existence of the empty set, as one thing, accords with the intuition that a singleton is a different thing to its element (although I would rather begin with a pair of definite things). But that there can be no set of all the pure sets is worse because we cannot resolve this paradox by pointing to the intrinsic fuzziness of (e.g.) the totality of the possible red chairs.
......The totality of all the pure sets is a pure proper class, a class being a collection of all and only those things satisfying some definite predicate, and a proper one being one that is not a set. Mathematicians who are interested in sets tend to regard mathematics as set theory, and so they can ignore proper classes, but the philosophical puzzle remains, motivating us to develop a plausible theory of classes—to point to the definitive property that sets have, and that classes lack. To begin with, if each pure set exists, as a definite thing, then all of them will, their totality existing as some sort of collection. And clearly, the totality of the pure sets and the totality of the cardinal numbers are two proper classes, and each has as many members as the other.
......In short, whatever intuitions led us to transfinite mathematics are likely to lead us towards trans-set mathematics, and inconsistency. In particular, if all the pure sets exist then not only would that totality seem to exist, each collection of pure sets would seem to exist already, as a sub-collection of that totality, whose power-class would therefore also seem to exist—much as the (standard full) power-set of the natural numbers seems to exist. So note that most of the subsets of the natural numbers (which collectively make that power-set larger than aleph-null) correspond to random sequences of 0s and 1s. Although the totality of the natural numbers is clearly a definite collection, a random sequence of 0s and 1s, which is like the probable result of endlessly tossing a fair coin, may well not be a definite thing, because its elements are not determined by a finite rule.
......If the natural numbers are like that (a potential infinity) then, collections being collections of things rather than of fuzzy stuff, numerical collections (at least) avoid Cantor’s paradox—there is only really a paradox when one has (with Cantor) the intuition that the natural numbers form an actual infinity. For Cantor, each potential infinity presupposed an actual infinity (since the former was for him like a variable ranging over the latter) so he thought of proper classes as being actual infinities that were nonetheless inconsistent, a view that is unattractive to most philosophers (who would conclude from an inconsistency the falsity of one or more presuppositions).
......So the problem is, if it is not the endlessness of the simply infinite sequence that makes some infinite collections (those whose members are not determined by any finite rule) unfit to be members of other collections, then what is it? (Note that even if the natural numbers are potentially infinite, their totality may nonetheless be regarded one thing, with a definite number of elements—the same as the number of the rational numbers, via the standard bijection between those two collections—although the devil is in the details).

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