Computing Exact Approximations of a Chaitin Omega Number: A Glimpse of Randomness

In this monograph,the research aimed to compute some exact bits of a Chaitin Omega number. A Chaitin Omega numbers are halting probabilities of a specific mathematical model of the ubiquitous PC called 'self- delimiting Turing machine'. In 1936,Turing showed that no mechanical procedure and therefore no formal axiomatic theory can solve Turing's halting problem, the question of whether a given computer program will eventually halt. An Omega number combines all instances of Turing's halting problem into a paradoxical real number. Its binary digits or bits are algorithmically random and cannot be distinguished from the the result of independent toss of a fair coin. Omega has a simple mathematical definition,but it does not enable us to determine more than finitely many of its digits and no other definition can do it better. Furthermore,as nobody before was able to compute any exact bit of a natural Omega number, the carrying on the computation is much more demanding than solving Turing's halting problem. We reviewed the properties of Omega numbers leading to the computation of approximations to obtain initial exact 64 bits of a Chaitin Omega number.