On Chaitin's Heuristic Principle and Halting Probability

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 number Omega, and investigate whether it is really a probability of halting the programs. We will see that the family of all the prefix-free binary string sets is not a sigma-algebra, and the supposed measure, on which Omega is defined as a halting probability, is not a probability. In particular, the non-halting probability is shown to be not 1--Omega. We will also suggest some methods for defining the halting probabilities.