I don't think that it's actually formulating a bijection across the reals - the generator function looks to me to be generating the complete set of rationals.

> Represent each real number x∈[0,1) as a sequence of DIGITS, where each DIGIT is either in {0,…,8} or is of the form 10∗(10k−1)+i with k≥1 and i∈{0,…,8} (i.e., in {90,…,98;990,…,998;9990,…,9998;…}.

He's iterating to infinity, that sure as shit sounds countable to me.

He's associating to each x in [0,1) an infinite sequence of natural numbers (just like decimal representations do), and there's uncountably many such sequences

I may have misread that one, but I still think the reverse mapping only gets as far as within arbitrary bounds.

But he's defining the numbers with a property that makes them countably infinite?

The part that seems problematic is "each real number" the reals exist across bounds, I'm not sure there can be an "each".

I think his iterator is generating infinitesimals of rational and not continuous nature.