Every Number is Describable?

Loosely, a number is describable if it can be unambiguously defined by a finite string over a finite alphabet. Numbers such as $\frac{1}{3}$, $\sum_{n=0}^\infty \frac{1}{n!}$, and "the ratio of circumference to diameter of a circle" are all describable numbers. One can show that the set of all such numbers is countable.

Let $U$ be the set of indescribable real numbers. I "claim" $U$ is empty, and so every real number is describable. Suppose to the contrary. Place a well-order on $U$ (this can be done assuming the Axiom of Choice). The least element $u$ of $U$ admits the description "the least element of $U$ according to the specified well-order", contradicting the indescribability of $u$.

Obviously, something has gone awry here. I suspect it is some combination of my loose definition of describability and my self-referential "description" of $u$. My (slightly open-ended) question is "What's wrong here?".

P.S. There is a blog post addressing something rather like my question. As I understand it, the author's response is "the given description isn't really a description". If so, I'd enjoy some elaboration.

Solution 1:

But you did not describe the well-ordering of $U$. You merely described the fact that it exists.

You extended your language, and this allowed you to describe another countable set of numbers. Alas repeating the argument until exhausting $U$ would require an uncountable number of iteration, and by that point the language is no longer finite.

Let me give a slightly more advanced analogy. One could argue that in $L$ (Godel's constructible universe) everything is definable including the well-ordering of $\mathbb R$. Recall that GCH holds in $L$ and for every countable $\alpha$ there are new real numbers added in $L_\alpha$, so the construction of the real number is not exhausted until $L_{\omega_1}$.

But in the construction of $L$ we allow parameters which were previously constructed to be used. This allows us to extend our language, so to speak, and by that to generate another countable set of real numbers. In order to exhaust the entire collection of real numbers we had to get to $L_{\omega_1}$, that is to say that we had to make uncountably many steps. So no countable iteration covered everything.

Solution 2:

Besides the issue with the axiom of choice possibly giving you an undefinable well ordering, there is still the usual issue that "unambiguously defined by a finite string over a finite alphabet" does not succeed in defining a set of real numbers. The actual set existence axiom in set theory only allows you to form a set of real numbers if the set is definable by a formula of set theory. The quoted phrase cannot, in fact, be expressed by a formula of set theory.

There is a very thorough answer by Joel David Hamkins located here, which explains the situation in more detail.