Chaitin Ω numbers, Solovay machines, and Gödel incompleteness

AbstractComputably enumerable (c.e.) reals can be coded by Chaitin machines through their halting probabilities. Tuning Solovay's construction of a Chaitin universal machine for which ZFC (if arithmetically sound) cannot determine any single bit of the binary expansion of its halting probability, we show that every c.e.~random real is the halting probability of a universal Chaitin machine for which ZFC cannot determine more than its initial block of 1 bits—as soon as you get a 0, it is all over. Finally, a constructive version of Chaitin information-theoretic incompleteness theorem is proven