## Monday, September 08, 2008

The positive integers (1, 2, 3, ...) are the products of endlessly reiterating the addition of 1, starting with 1. Given a common sense arithmetical realism, either they exist (so to speak) altogether like stars coexisting in space, forming a transfinite set—a quasi-spatial collection that is a definite thing in its own right—or else they are more accurately envisaged quasi-temporally, being never a completed collection (or set) but rather being (in Mill’s words) indefinitely extensible. It has become standard to suppose the former, so let us do that, for now.
......For any set there is also its power-set, the set whose members are the subsets (the parts) of the original set. And each set is (cardinally) smaller than its power-set because not only is the former a subset of the latter, there is no way to pair the members of the former with the members of the latter (the proof of that impossibility is basically Cantor’s diagonal argument, which uses axioms that are realistic enough). Reiterating the power-set operation, starting with the set of positive integers, yields an endless sequence of transfinite sets, whose (cardinal) sizes are the Beths. Now, if there was a set of all the other sets, its power-set would contain more sets, which is absurd, whence there is no such set. The sets—and similarly (although the argument is longer) the Beths—therefore form a different sort of collection, a proper class.
......That result is Cantor’s Paradox (which I last blogged about over a year ago). Each integer follows from some finite reiteration of their defining (and clearly definite) algorithm, so it exists, so to speak; they all do, whence one might expect that they form a set. And since that set exists (quasi-spatially), so each proper part of it exists (as a different collection), whence the power-set whose size is the next Beth exists; and so on. Each Beth results from a definite transfinite reiteration, so they all exist. Cantor’s paradox is so-called because realistic intuitions that the integers form a set tend to give the wrong answer for proper classes such as the Beths.
......Consequently those intuitions are not to be trusted; and indeed, there is also an empirical argument—from an equally realistic interpretation of quantum-mechanical probabilities—to the indefinite extensibility of applicable arithmetic. Mathematicians have for the most part preferred to reject such realisms, rather than (formalistic) set theory, presumably because of the predominant Naturalism within academic Science. But Cantor was himself a realist, and not afraid of theism, and might have rejected his set theory before either. (His way of keeping all three was to go paraconsistent when it came to theism, but that did not really solve the problem for arithmetical realism.)