The construction counts an arbitrary length list of numbers, each which can count an integer > 0. The case between a number < 1 and a number >=1 is also addressed.

If that holds true without missing any possibilities, then this maps to the continued fractions. I've gone through several such mappings that missed at least one before, so looking to see if this one has any such as well.

https://math.libretexts.org/Bookshelves/Combinatorics_and_Di... A bit before and after section (6.1.13) covers how the continued fractions then provide a counting of the reals, provided any finite representations are > 2 for the last number.

So well I'm not sure it all holds up, it does appear that way.