I don't see how downward L-S would let you name all the reals. I don't think anyone claims you can name all the reals, L-S or not. Perhaps I'm wrong. But I'm generally familiar with the reference you linked and I don't see how it implies that you can name all the reals. If there are names and/or algorithms for all the reals, that would pretty much demolish the entire field of algorithmic complexity theory. Nor would anyone have any further need to care about computable or definable sets of numbers.