It seems to me that Cantor's diagonalization fails here because of the very different nature of descriptions vs (for example) decimal notation. Every possible string of digits is a valid, unique number. That does not apply to descriptions.
I'd assume that every number that can be precisely described by some symbolic notation can be described in that notation in multiple ways, and likely in an infinite number of multiple different ways. E.g. the number 2 can be described as 1+1, 1+1+1-1, 1+1+1+1-1-1, ad infinitum.
Furthermore, I'd assume that not every string in that symbolic notation constitutes a valid, precise description of some real.
So Cantor's diagonalization produces some unique description of a number that differs from all of the descriptions - but it's possible and plausible that the description refers to a number that is in the list but has been described differently; and it's possible and plausible that the constructed description does not describe any real whatsoever.
Or am I completely misunderstanding you and you did not intend to apply Cantor's diagonalization to the descriptions?
I don't mean to apply the diagonalisation procedure to the descriptions. That wouldn't work for the reason you mentioned, and also because applying Cantor's diagonalisation to a bunch of finite strings might yield an infinite string.
What I meant was to apply Cantor's diagonalisation to the decimal expansions of the describable numbers. Take all of the describable numbers ordered lexicographically by their lexicographically first description, and then look at their decimal expansions and describe a new number that differs from the nth one in the nth decimal place (with the usual details to make sure you don't end up with a second representation of a number already present).
This gives the decimal expansion of an alegedly indescribable number, because it's different from all the ones on the list. But because I can describe the diagonalisation procedure, this decimal expansion is itself a valid description, and hence we have a contradiction.
I'd assume that every number that can be precisely described by some symbolic notation can be described in that notation in multiple ways, and likely in an infinite number of multiple different ways. E.g. the number 2 can be described as 1+1, 1+1+1-1, 1+1+1+1-1-1, ad infinitum.
Furthermore, I'd assume that not every string in that symbolic notation constitutes a valid, precise description of some real.
So Cantor's diagonalization produces some unique description of a number that differs from all of the descriptions - but it's possible and plausible that the description refers to a number that is in the list but has been described differently; and it's possible and plausible that the constructed description does not describe any real whatsoever.
Or am I completely misunderstanding you and you did not intend to apply Cantor's diagonalization to the descriptions?