User blog comment:PsiCubed2/An FS-independent injection from the Countable Ordinals to the reals? (help me find the catch)/@comment-5529393-20171219192605

The catch is that there are many reals that represnt the same ordinal. You can indeed define a one-to-many mapping from the countable ordinals to the reals, and indeed a surjection from the reals to the countable ordinals. But, you cannot define an explicit injection from the countable ordinals to the reals, because it is consistent with ZF that there isn't one.