Ready.
Let me tell you a "secret." I Am Not Convinced with the proofs (those so far I have read/processed) about the uncountability of all of the strings. Let's find out what is being misunderstood on which side.
In 1994, I told a theory-of-computation lecturer that I had a disproof, he told me "Tell it to me. You get an AA, I get the Nobel." (To correct tha first, ACM's Turing, or the mathematical Hardy awards may be more relevant. They are not awarding Nobels and/or Oscar, Tommy, Emmy, etc. for such work. (Taking it as a joke/poke, I am skþppþng the ethical question with the suggestion.))
The problem, as I look at my old jotted notes is that, the exact ratio is definable: It is (2**n)/n. i.e:The nth power of 2, divided by n.
I may later be presenting also an example definition to map the (2**n)numbers to n numbers, just as in the case of mapping (n*n) to n. This may be my lack of formal language, beyond the references I have, but given that none of them even include this formula, suggests me that the proof may be shakable. Or, that, I am hinting to a better proof, if such a mapping is not possible. (e.g. Starting with only one-bit numbers all the rest of infinite bits being 0, then adding two-bit numbers (except zero) likewise to the set, then 3-bit numbers, and so on, upto n-1, for each n. Recursively definable. Right?)
In my jots, in one of the books, (on p.10 of J.Brrokshear's 1989 book from Benjamin-Cummings)
This (the correspondence ratio of the range) is on the order of (n*(n-1)). lim (n*(n-1))/n goes to infinity as n goes to infinity. Likewise, for (2**n)/n.
I an ready to accept an argument of increasingly less-countableness., if one such were made, but I reject inclusion of one and the exclusion of the other as countable sets. (this sounds very much as a "paradox of 'definability'".)
In other words, the behavior is formulatable. Yes, it is an exponential-mapping (2 to the power of n) versus n-squared. But, nobody states it as such - at least the references I have. And as a result, the proof is not given on those terms.
The question is: Do we take that as a qualitative difference, as opposed to a quantitative?