  It would be a heavenly reward if there were a method of weighing theories and
sentences in such a way that a theory could never prove a heavier sentence
(Chaitin's Heuristic Principle). Alas, no satisfactory measure has been found
so far, and this dream seemed too good to ever come true. In the first part of
this paper, we attempt to revive Chaitin's lost paradise of heuristic principle
as much as logic allows. In the second part, which is a joint work with M.
Jalilvand and B. Nikzad, we study Chaitin's well-known constant Omega, and show
that this number is not a probability of halting the randomly chosen input-free
programs under any infinite discrete measure. We suggest some methods for
defining the halting probabilities by various measures.
