Turing proved the existence of non-computable reals, so the two are not the same. Also Chaitin's constant is a non-computable real number with a precise mathematical definition.
See: http://en.wikipedia.org/wiki/Chaitins_constant
This means that it is possible to express the proofs of your statements in code.
On the other hand, non-halting programs are AFAIK beyond the reach of formal logic, and as consequence, the set of logical proofs (or more accurately, its isomorphic twin) is a subset of the set of all computer programs.