That's how my Set Theory prof described a few of the results in today's class. And, they are. Once one admits that a set can be so large that it can not even be enumerated, much less bounded, well, crazy things can happen. And, that's all good. As I've said before, sending your mind out to do battle with concepts that can't possibly be grasped is a big part of math's appeal.
But, it's also important to recognize that math is our invention. When we "prove" something in math, we're not really proving anything other than that a certain assertion is consistent with a certain set of assumptions that we made up. There's no real "truth" going on here.
The axiom of choice is just one of those things that sounds like it ought to be true, so we say that it is. If nutty results follow from that, it could be that some things are just nutty, but it could also be that the axiom of choice is nutty. Particle physicists are increasingly questioning if something as generally accepted as the "real" numbers even exists. The empirical evidence is certainly consistent with a discrete and countable version of the universe. Sure, that means that something has to be able to move from one point to another without passing through any intermediate points, but why is that such a stretch? Anybody who's ever seen a movie projector should know that it's pretty easy to trick our lame senses into assuming continuity when no such thing exists.
I'm not ready to write off the continuity of the reals quite yet, but I'll at least be man enough to admit that this whole edifice is built on sand.
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