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

    prefix = '0.';
    queue = new Queue(prefix);
    for(;;){
        s = queue.poll();
        for(c:['0'...'9']){
            queue.add(s + c);
        }
    }

When the constraint is "run forever" I don't think that the generator he defined produces results any different from the snippet above.

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.

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