Not in Kansas anymore

As I read the following proof in my Set Theory text, I was reminded of Dorothy's line to Toto. I won't even state what's being proven, because the weirdness of the proof is greater without context:

Obviously, e ≤ d + e. By Theorem 9 it will suffice to prove that d + e ≤ e. Now d + e ≤ e + e, and by Theorem 13, e + e = e.

OK, if you were to throw that at a High School kid, they'd give you the first sentence and then say you're full of crap. An industrious Undergrad might conclude that d and e are both zero. In fact, the opposite is true. We have left the familiar realm of finite numbers and are now dealing with different levels of infinity: the cardinal numbers. It's wierd stuff, but again, you just go with it and it all works.

Incidentally, if you came away from my previous post on the axiom of choice thinking that you can just dismiss this stuff as mathematical musings, you missed my point. Yes, these are human constructs and nature doesn't have to play that way (and it's quite possible that it doesn't). But that doesn't make them irrelevant, either.

Consider the Naturals, {1, 2, 3, ...}. Everybody agrees that the Naturals are just that, natural. There's even a famous (by math standards) quote: "God created the Natural numbers, everything else is the work of man." by Leopold Kronecker (1823–1891). The point is, counting is as real as math gets. So, what happens if you take the power set of the naturals, that is every possible subset of them. How many subsets of the Naturals are there? Hells bells, the cardinality of that set is c, the continuum. The same as the Real line. All you've done is acknowledge that you might want to look at subsets of the most obvious thing out there and you're already into uncountable sets and all the craziness that comes with them.