Saturday, December 11, 2010

Cantor’s Paradox cont.

This is the fourteenth of 17 posts, which are collectively Eternity, etc.
......You may be familiar with N (see previous post) from school mathematics. Such informal sets are basically collections that are quasi-spatial, in the sense that their members coexist (insofar as they do exist) altogether. Given any spatial collection—e.g. some ordinary objects in a room—any sub-collection of them is clearly also spatial; and similarly, a definitive property of our informal sets is that every conceivable sub-collection of such a set is itself quasi-spatial, is a subset [i].
......Surprisingly, the modern view (of arithmetic as timeless) offers little support to atemporalism, as follows. To say that two collections have the same cardinality—the same cardinal number of members—is to say that the members of each collection could all be paired off, one-to-one, with those of the other [ii]. And for any set, S, if the collection of all its subsets is also quasi-spatial, then that collection—including (for simplicity) the so-called improper subsets, S and the empty set—is the powerset of S, say P(S). And according to Cantor’s Diagonal Argument [iii], P(S) always has a greater cardinality than S.
......In particular, the cardinality of P(N)—which Peirce called “Beth-1”—is greater than the cardinality of N, which is Beth-0. And the cardinality of P(P(N)) = P-squared(N) is Beth-2, which is greater than Beth-1. And so on; for each natural number n, P-to-the-nth-power(N) has Beth-n members. And the union of N and all those P-to-the-nth-power(N) is the collection of all their members. For each n it contains at least Beth-(n + 1) members. So its cardinality, say Beth-omega [iv], is greater than Beth-n for every n. And if that union is also a set, say U, then by Cantor’s Diagonal Argument, P(U) has an even greater cardinality, Beth-(omega + 1) [v],
......We might expect that union to be a set, because Beth-0 being an Actual Infinite number means that all those Beth-0 sets coexist quasi-spatially (like a row of houses, whose contents therefore coexist similarly). So the next union might be of U and all the P-to-the-nth-power(U). But by continuing in that way, taking powersets and unions as far as is logically possible [vi], we cannot end up with a set because from any set we could have continued further in that way. We have, then, a collection that is not quasi-spatial, being generated by a process that cannot be completed (as a matter of logical necessity). Continued...
......Notes:
......[i] By contrast, if the natural numbers are forever growing, according to the rule of add 1 repeatedly, then only those sub-collections that are similarly specified by a finite rule exist in the same kind of way.
......[ii] The natural numbers are finite cardinal numbers. And N has the same infinite cardinality as the subset of just the even numbers because n can be paired with 2n for all natural numbers n. There are, in an obvious sense, more natural numbers than even numbers, but cardinality is fundamental to our number concept; Shapiro, Thinking about Mathematics, pp. 133–8.
......[iii] If S and P(S) had the same cardinality, there would be one-to-one mappings from S onto all of P(S). Let M be one such mapping, and let a subset of S, say D, be specified as follows: For each member of S, if the subset that M maps it to contains it then D does not contain it, and otherwise D does. The problem is that since D differs from every subset that M maps the members of S to, D differs from every subset of S, whereas D is by definition a subset of S. That is, D is contradictory, and so there is no such M, which means that S and P(S) do not have the same cardinality. But for each member of S, say m, P(S) contains {m}, and so the cardinality of P(S) is greater than that of S.
......[iv] Omega is the first ordinal number after the natural numbers. Ordinal numbers generalize counting numbers as such beyond the natural numbers (whence their use indexing the Beths).
......[v] Such ordinal addition corresponds to a rearrangement of the natural numbers, e.g. from their natural order (to which omega corresponds) to 2, 3, ..., 1.
......[vi] We could also take unions of Beth-1 sets, since Beth-1 is Actual Infinite; and similarly, Beth-2 sets, etc.

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