I know [Chaitin's constant](http://en.wikipedia.org/wiki/Chaitin%2527s_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](http://books.google.com/books?id=tRdoIhHh3moC&printsec=frontcover&dq=klee+and+wagon&source=bl&ots=BwPb3iorQw&sig=oqbPFWeRYfPcbKnAox_Lvx_wdfY&hl=en&ei=RK8BTO7gB4O0lQe7_OSiCA&sa=X&oi=book_result&ct=result&resnum=1&ved=0CBYQ6AEwAA#v=onepage&q&f=false) 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](http://rjlipton.wordpress.com/2012/06/15/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.



{transcendental numbers} \ {computable transcendental numbers}
Note: Answer is pending update per attached comments.


The difference, stated informally, is that that the non-computable transcendentals in their k-base digit representation are entirely random and non compressible. A computable transcendental, such as e, can be represented by a finite algorithmic description, such as a series expansion, which is a form of compression. For the non-computable numbers no such shorter representation exist. Their shortest computational description is their own infinite digit sequence.
You can read more about computational complexity here:
<http://en.wikipedia.org/wiki/Kolmogorov_complexity>.


There is a wealth of similar numbers to the Ω class of numbers. In general it is "easy" to come up with new definitions for such numbers. These all belong to the countably infinite set of non-computable definable numbers.


To make matters worse, what is left are the non-definable (and therefore also non-computable) numbers. They are the numbers that cannot be described in any way what-so-ever, other than by just iterating through their infinite non-compressible digit sequence. The set of all non-definable numbers is uncountable.


