stoutfellow

At least to begin with, Georg Cantor was just interested in the behavior of Fourier series. His studies, though, were to lead to a thorough restructuring of the foundations of mathematics, and touch off a civil war (not over personal pique, as in the case of Newton/Leibniz, but over fundamental issues concerning the nature of mathematics) that still sputters along to this day.

One of the questions that arose in connection with Fourier series was that of uniqueness: to what extent was it possible for two different Fourier series to describe the same function? Equivalently, if a Fourier series evaluates to zero "most of the time", must it actually be zero - i.e., must its coefficients all be zero? It was shown early on that if the series always evaluates to zero, then it actually is zero. Cantor's early work tried to extend this a bit; he eventually showed that if a Fourier series isn't zero, then the set of points on which it isn't zero is (in his terminology) "of the second kind".

This will take a little explaining. Let P be a set of real numbers. The "derived set" P' is the set of "accumulation points" of P - all those real numbers which have infinitely many members of P close by.

More precisely: let x be any real number. x is an accumulation point of P if, for every positive ε, there are infinitely many members of P within ε of x.

For example, if P is a finite set, P' is empty. If P={1,1/2,1/3,1/4,...}, then P'={0}. If P consists of all the rational numbers, then P' includes all of the real numbers. Now, you can take the derived set P'' of P', and the derived set P''' of P'', and so on. P is "of the first kind" if one of these successive derived sets is empty, and "of the second kind" otherwise. (The first two examples I gave are of the first kind; the third is of the second kind.)

What got Cantor into trouble was what he did next. Having generated derived sets P⁽ⁿ⁾ for every positive integer n, he suggested looking at the intersection of all of them, which he called P^(ω) - the ω being meant to suggest infinity. This set, in turn, would have a derived set P^(ω+1), and we're off: P^(ω+2), P^(ω+3), ...; again we take an intersection and get P^(2ω).

You can see where this is going. The superscript could become any multiple of ω ; it could - eventually - be ω²; there seemed no end, finite or infinite, to the succession of derived sets.

What's of great importance here is not the derived sets, but the superscripts, which Cantor dubbed "transfinite ordinals", each describing an "ordered set" - a set whose members are, so to speak, lined up. The finite ordinal 2, for example, describes a pair of objects, one behind the other. The ordinal ω describes the entire set of positive whole numbers. What do the new, transfinite ordinals describe? Well... to give another example, consider the set of pairs of positive whole numbers (a,b). Order them "lexicographically"; that is, (a,b) comes before (c,d) if a < c or if a = c and b < d. (Think of a,b,c,d as letters of the alphabet, and think of dictionary order.) This set is described by ω² - you have infinitely many copies of ω, all lined up one after the other.

This was Cantor's innovation: he began thinking about infinity, distinguishing between different infinities, and performing what looked like arithmetic operations on them. Long tradition, dating back to Aristotle at the least, rejected the notion of actually infinite sets, admitting only the virtually infinite. Galileo had poked at the edges of this; it was Cantor who actually took the plunge. There was more to come.

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