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.
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.