The Infinite Richness of Mathematics
In Noesis #136 Robert Dick raised the issue of orders of infinity, and presented the proof of Cantor's theorem that the power set of any set has a strictly greater cardinality than the original set. Cantor died insane, but along the way did much to put set theory on an axiomatic foundation. (Not to mention the fact that the diagonalization structure of that proof crops up over and over again, for example in proofs about computability, in Goedel's theorem, and so on.)
There are many interesting issues arising out of this, such as the fact that counting turns out to be a much more complicated notion for infinite sets than for finite ones: much of it depends on the exact axioms one uses for sets, and it is possible to construct set theories in which Cantor's theorem is false (and in which there is indeed a set of all sets); there is a theorem which states that any set of axioms has a countable model (and also a theorem that the real numbers are uncountable--this happy state of afffairs is called the Lowenheim-Skolem paradox).
But I don't want to talk about any of that highbrow stuff here, as it's all in books anyway. (Anybody who doesn't already know about it but wants to could do far worse than look between the covers of Mendelson's Introduction to Mathematical Logic, now in its 3rd edition.)
In fact, I want to talk about something considerably more elementary, which Robert's article reminded me of, and which was one of the mathematical gems which got me seriously interested in mathematics, way back when I was about 17.
First, we note that the rationals are a countable set. It's easy to form a list which will include any positive rational, as follows:
1 2 1/2 2/2 3 1/3 2/3 3/3 4 1/4 2/4 3/4 4/4 . . .
You can then go through the list striking out any rational number which has already appeared. This gives an enumeration of the positive rationals, since any positive rational p/q will appear somewhere in the list. You don't need to do this, but it makes things a little tidier.
This fact, that the rationals are countable, is the first surprising thing--after all, any number can be approximated arbitrarily closely by a rational number, so the rationals are dense in the reals. But there are certainly uncountably many reals.
But here comes the second, and more surprising, thing. Pick a positive number, say epsilon, and let rn be the nth rational number in the enumeration described above. Now, define the subset of the real numbers In by
In = those numbers x lying between rn - epsilon/2(n+1) and rn + epsilon/2(n+1)Then the length of In is epsilon/2n. But now, the total length of all the In's is just the sum of epsilon/2n with n going from 1 to infinity, which is epsilon. Now, the whole collection of In's includes all the rational numbers, and has total length at most epsilon (there may be overlaps, but that gives an overestimate of the total length). Since epsilon was arbitrary, the total length of the real line occupied by the rationals is less than any positive real number, and so must be 0.
Hence, the rational numbers--although they are dense--take up "no" space in the real numbers. For a real brain-bending exercise, try to visualize how you can find a sequence of sets like the In which includes all the rationals but somehow manages to miss almost all the reals.
I think it was this that gave me my first hazy glimpse of the immense richness of mathematics, even at the relatively elementary level of analysis I had access to at that stage.
I'd be interested to hear if any of those reading had similar experiences with mathematical proofs or results, whatever level they occurred at.
Rob. [E-mail address omitted.]