ihm's response to you I think did a great job explaining what I meant to say, but let me elaborate further.
Just like you might define computable real numbers, you may similarly describe the "definable" real numbers, that is, those numbers which are uniquely specified by logical statements. You will inevitably find that the definable real numbers, like the computable reals, appear to be countable.
So the fact that something is computable, per se, isn't exactly what makes the real numbers countable rather than uncountable.
I use the vocabulary of meta-theory and models, because the practice of defining either the countable reals or the definable reals within a formal system looks a whole lot like defining a formal system within itself (i.e., metatheory). So the fact that the computable reals and the definable reals appear countable, is, at least to me, much like the statement that there are countable models of your favorite formal system.
Just like you might define computable real numbers, you may similarly describe the "definable" real numbers, that is, those numbers which are uniquely specified by logical statements. You will inevitably find that the definable real numbers, like the computable reals, appear to be countable.
So the fact that something is computable, per se, isn't exactly what makes the real numbers countable rather than uncountable.
I use the vocabulary of meta-theory and models, because the practice of defining either the countable reals or the definable reals within a formal system looks a whole lot like defining a formal system within itself (i.e., metatheory). So the fact that the computable reals and the definable reals appear countable, is, at least to me, much like the statement that there are countable models of your favorite formal system.