Is there a link between uncountable sets and infinite information? There are only countably many things you can express with a finite number of words. This implies that any uncountable set has to contain uncountably many elements which you cannot define by any finite means. Each of those elements must have an infinite definition that you cannot compress to something finite.
Note that this is not just about countability of computable numbers. Even the most exotic numbers we can think of (e.g. Chaitin's constant) are countable because their definition is finite.
Is this reasoning correct?
 A: One issue with this is with the word "description"/"definition". What counts as a valid definition of a number? Consider the following: "The smallest positive integer not definable in under a million words." Is that a valid definition? If not, what goes wrong?
Suppose we run Cantor's argument on the definable reals, as follows. List all valid definitions of reals in order of length (use alphabetical order for definitions of the same length). Consider the real whose Nth digit is a 0 if the Nth digit of the Nth definable real is 1, and a 1 otherwise. Is this definable? Have I just defined it? (The issue with this is precisely the same as in my previous comment.) For more, look up the Berry paradox and Richard's paradox…
as well as Wikipedia's page on definable numbers.
Short resolution is: while we can define increasingly strong notions of definability, there is no ultimate notion of definability. If you hand me a definition system, I can create a stronger one that can define more numbers. If $D$ is a definition system, then the phrase "The smallest integer not definable in $D$ under a million words" is not a definition in $D$, but perhaps is a valid definition in a larger system $D'$. Each definition system on its own can only define countably many numbers, though.
