[ionfish]

The usual way this is done is by selecting a canonical representation for the reals in the list. It's pretty much the same thing as you said, but I don't think you're putting it in the right terms.

The list you're diagonalising (in Cantor's diagonal argument that the reals are uncountable) is a list of real numbers. Each of those reals has a canonical representation, and the reals in the list are ordered according to an order defined on those representations. The argument then demonstrates a method of constructing from those representations a representation of a new real which, by definition, cannot be one of the reals in that list.

Put in those terms it feels a lot less arbitrary than simply excluding certain strings.

[ColinWright]

Problem there is that you then have to prove that the string you've constructed is in your canonical form, and it might not be.