Cantor's diagonalization theorem, which proves that the reals are uncountable, is a study in contrasts. On the one hand, there is no question that it is correct. On the other hand, not only is it controversial, it attracts an inordinate number of cranks. Asaf Karagila compiles an excellent list of various "cranky" questions here.
I have come to the conclusion that math populizers have incorrectly concluded that the proof of Cantor diagonalization is accessible. For example, see this in the popular business press. My view is that all such "populist" proofs assume way too many sophisticated mathematical proofs to be viable.
Specifically, the standard Cantor proof assumes, without statement, that:
- countable sets are enumerable; i.e. can be put into a sequence
- sequences are valid ways to define new numbers (limits, sums), despite the fact that they contain an infinite number of terms
- the constructed real not in the list of "countable" reals is the limit of a Cauchy sequence, and is therefore a real number and not rational (which explicitly references the completeness of the reals)
So my question is on either side of my examples:
- Given my protestation, is the standard proof by contradiction salvageable for non-mathematicians; and
- What amount of rigorous mathematics is needed to make this proof "crank proof"? I would say what I have provided is sufficient, realizing that serious cranks are impervious to anything not in their thinking!
EDIT: Thanks for the answers. I think I have a good idea where the line of (mis)understanding might be. Multiple people have expressed the notion that non-mathematicians understand the real numbers as infinite decimals; i.e. through their standard notation. This is where I think we mathematicians are wrong, at least for some people. I think lots of people doubt the existence of any such infinite object which seem to require an infinite construction, like Cantor. Specific reals, like $\pi$, have other definitions that seem "non-infinite". I accept that those who cannot grasp such infinite constructions may be a lost cause for popularization, but I do think that acknowledging this gap is important.