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.

翻译：若能存在一种权衡理论与语句的方法，使得一个理论永远无法证明一个“更重”的语句（柴廷启发式原理），那将是天赐的奖赏。然而，迄今为止尚未找到令人满意的度量方法，这一梦想似乎美好得永远无法实现。本文第一部分试图在逻辑允许的范围内，尽可能复兴柴廷那已失落的启发式原理乐园。第二部分是与 M. Jalilvand 和 B. Nikzad 的合作成果，我们研究了柴廷著名的常数 Omega，并证明该数字在任何无限离散测度下都不是随机选取的无输入程序的停机概率。我们提出了通过不同测度定义停机概率的若干方法。