Abstract:
Computably 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's all over. Finally,
a constructive version of Chaitin information-theoretic incompleteness theorem is
proven.