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 ever to 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 several methods for defining halting probabilities using various measures.

翻译：若存在一种权衡理论与语句的方法，使得理论永远无法证明更重的语句（蔡廷启发原则），那将是天赐之福。然而，至今尚未找到令人满意的度量，这一梦想似乎过于美好而无法实现。本文第一部分试图在逻辑允许的范围内，尽可能复兴蔡廷失去的启发原则天堂。第二部分（与M. Jalilvand和B. Nikzad合作）研究了蔡廷著名的常数Ω，并证明该数在任何无限离散测度下，都不是随机选择无输入程序停机的概率。我们提出了几种利用不同测度定义停机概率的方法。