Real Numbers as Well Defined Sets For every construction of the reals, we define a real number to be some kind of set of rational numbers (such as cuts or sequences).
However, the number of symbols we have to formulate the description of a set is finite (analytic functions, infinite sums, etc), and I assume sets can only be defined as finite strings of logical symbols.
Every number I can think of, including transcendentals, and even the Chaitin Constant, can be described as such.
Where is the fallacy in saying, then, that the real numbers are countable (as, for instance, well defined cuts seem to be)? 

The diagonalization argument will then say that any describable sequence of describable real numbers is not exhaustive, because there is a process for converting the description of the sequence to a description of a real number that is not in the sequence.    
The collection of all describable reals can be countable or not, and might or might not form a well defined mathematical object, depending on definitions and on the framework in which it is considered (is your mathematics describable-ist or not?). What diagonalization shows is that it is never describably countable.
