Chaitin's constant

A Chaitin constant, also called a Chaitin omega number or halting probability, is a real number from algorithmic information theory that, informally speaking, represents the probability that a randomly constructed program will halt. The numbers are formed from a construction due to Gregory Chaitin. There are infinitely many halting probabilities, one for each universal method of encoding programs, but it is common to use the letter Omega as if there were only one, and because the value depends on the encoding it is sometimes called Chaitin's construction. Each halting probability is a normal and transcendental real number that is not computable, so no algorithm can compute its digits, and it is Martin-Löf random, so no algorithm can even reliably guess its digits. Many outstanding problems in number theory, such as Goldbach's conjecture, are equivalent to the halting problem for special programs, so knowing enough bits of the constant would answer them, but calculating more than the first few bits is not possible for a universal language.