TITLE: Example of uncomputable but definable number
QUESTION [38 upvotes]: Every computable number is definable. However, the converse is not true.
What is an example of a real number that is definable but that is NOT computable? I guess if it is there, we can "define" (describe) it, can't we?

REPLY [3 votes]: The simplest is perhaps the uncomputable real number whose binary expansion is 
$$0.x_1x_2x_3...$$
where
$$x_i = 
\begin{cases}
1 & \text{if }T_i\text{ eventually halts}\\
0 & \text{otherwise}
\end{cases}
$$
and $T_i$ is the $i$th Turing machine (in some chosen ordering) with an initially blank tape.