If any set that you can describe in a finite language is countable, then how is R not countable?

No.12009003 ViewReplyOriginalReport
There aremany very short (and pathetically simple) sequences of numbers and letters that describe the set of all real numbers. For example, the 11-symbol string "Dedekindcut" describes all real numbers, proving that R is extremely countable.

Is this because Dedekind cuts fail to reach all real numbers, or is it because R itself is actually countable?