......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-

*n*th-power(N) has Beth-

*n*members. And the

*union*of N and all those P-to-the-

*n*th-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-

*n*th-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 2

*n*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.

## No comments:

Post a Comment