> No one should take what the good Dr. Ben Zona writes too seriously. > But there is an important question that comes from it: One comes up > with a paradox when one defines log (aleph_0). > > So can anyone rigorously show that there is no possibility of also > arriving at a paradox when one defines the irrational number log_2 3? > What makes it OK to do this here when it is clearly not OK to do this > with log(aleph_0)? > > Here is another question: I look on the web and find a ton of papers > trying to refute Cantor's Theorem. (Just Google "Cantor diagonal > wrong" and see what you get.) I have never seen one of them that > convinced me. Why is there so much opposition to this theorem when > most other famous theorems are pretty much universally believed?
I think this is a good question. I don't know the answer, but I suspect it runs something like this.
Cantor's theorem is pretty much the simplest proof of an unintuitive result that I know. It doesn't require much background before one can be shown the proof. Thus, it is often a student's first exposure to a surprising theorem and one which conflicts with naive intuitions.
Those that become cranks work hard to reject this result. Those that proceed in normal ways in the mathematical curriculum accept this result (while, perhaps, emphasizing a distinction between "size" and "cardinality" to keep their intuitions intact). As they see other counterintuitive but valid results, it begins to bother them less and less. Mathematics is about deductive consequence, and if the consequences of our axioms are counterintuitive or surprising, well, that's too bad for our intuitions (or perhaps we must in extreme cases reconsider our axioms).
The student that gets all the way to the Banach-Tarski paradox is unlikely to argue against the result. If he follows the proof, then he accepts that it is a theorem of ZFC. Perhaps some will find this a reason to reject choice, but giving an argument against a certain axiom is not necessarily crankish.
Footnotes:  Some students find it implausible that there are infinities of different "size", others find it implausible that the set Q has the same "size" as N. You'll find cranks that argue mostly against the first result, and others that argue mostly against the second and some that argue equally fervently against *both* results. The most reasonable of this last group don't argue that both results are wrong exactly, but that the axiom of infinity yields contradictions.
 I remember very well that I did this when I learned Cantor's theorem. I thought it was all very odd that there were different sizes of infinity, so I consoled myself that it was a purely technical result about a technical term, namely "cardinality". It's not *really* about sizes, I figgered, so I slept well that night.
-- Jesse F. Hughes "Well, I don't claim to be an expert, in fact I am a fry cook with a national burger chain, but I have solved many differential and partial differential equations numerically." --C. Bond