I've never heard of this. Can you provide more detail?
It seems to me that it's unquestionably the case that there are only countably many finite-length strings over a countable language. This applies to English as well as any logical system you might devise. Since there are uncountably many reals, it clear we can only name or algorithmise (if that's a word) countably many of them.
This reasoning would apply to the language or the metalanguage, whatever that means. There simply aren't enough finite-length strings to go around.
Of course if you allow infinite-length strings then any real can be named by its decimal expansion. But infinite-length strings defeat the intuitive meaning of an algorithm.
Consider a countable model of ZFC, or a countable elementary submodel of your model-of-choice-for-sets. (this exists by Skolen_Lowenheim)
Then you can consider sets as labelled by natural numbers, and then you have 1-1 correspondence between the sets of algorithms (as that term means in the model) to the set of reals (again, reals that are in your submodel). This is the basis of Skolem's paradox.