I know Chaitin's constant Ω is not computable (and therefore transcendental). Are there other specific, known noncomputable numbers? I am trying to understand what distinguishes a computable transcendental number, such as π, from a noncomputable transcendental number, such as Ω. Is there anything revealing that can be said about the set difference {transcendental numbers} \ {computable transcendental numbers}?

I ask this as a novice. I am re-visiting a wonderful book that sadly can no longer be updated by Victor Klee, in which he and Wagon pose this as an open problem: *If an irrational number is real-time computable, is it then necessarily transcendent?* [Problem 23]

**Update** (*19Jun12*). There is an illuminating discussion under the title
"Why The Hartmanis-Stearns Conjecture Is Still Open" at the Lipton-Regan blog.
The Hartmanis-Stearns Conjecture is the open problem mentioned above:
If a number is real-time computable, it is either rational or transcendental.
If true, this has what strikes me as a counterintuitive consequence: that algebraic irrationals like $\sqrt{3}$
are in some sense "more complicated" than transcendentals.

nota specific number. It is only defined relative to a universal prefix-free machine, and there are many choices of such a machine with no single "correct" choice. Any sense of "specific" that applies to Ω also applies to the Turing jump of the empty set, 0'. – Carl Mummert May 30 '10 at 2:47