[baddox]

What do you mean by "represented with a finite amount of information"? Are you referring to their representation in a positional notation like decimal or binary? Or are you referring to the much subtler and more advanced fact that almost all reals are uncomputable? The former isn't really true, and the latter, while true, is subtle enough that it doesn't matter for the vast majority of mathematics (and to replace the reals with the computable numbers would make most of mathematics messy).

[gizmo686]

I don't think "represented with a finite amount of information" means computable. For example, consider BusyBeaver(n). We have shown that their exists an n such that BusyBeaver(n) is uncomputable. However, "BusyBeaver(n)" still contains enough information to describe this number. However, because all descriptions are a finite string from a finite alphabet, we can show that only a countable infinity of descriptions exist. However there exist an uncountable infinity of real numbers. Therefore, most real numbers cannot be unambiguously described.

[baddox]

Again, it just comes down to what we mean by "information" and "description." We can certainly construct the real numbers using a finite amount of precise language, so it's reasonable to claim that we have described all real numbers. Heck, even the existence of the English phrase "all undescribable real numbers" evokes an interesting linguistic and philosophical debate, similar to http://en.wikipedia.org/wiki/Interesting_number_paradox.

[gizmo686]

We can construct the set of all real numbers with a finite amount of information. However, that set contains elements which we cannot precisely describe with a finite amount of information.

The phrase "all undescribable real numbers" does not introduce any problems, because we have still not described any specific undescribable number. We would run into a problem with a phrase such as "the smallest undescribable real number", as that would be a description of a specific undescribable real number. Fourtuantly, that particular phrase does not raise any problems because we can simply conclude that their is no smallest undescribable real number, in the same way that there is no smallest real number in general.