Abstract:
A real is computable if it is the limit of a computable, increasing, computably converging sequence
of rationals. Omitting the restriction that the sequence converges computably we arrive at the notion
of computably enumerable (c.e.) real, that is, the limit of a computable, increasing, converging
sequence of rationals. A real is random if its binary expansion is a random sequence (equivalently,
if its expansion in base b ≥ 2 is random). The aim of this paper is to review some recent results on
computable, c.e. and random reals. In particular, we will present a complete characterization of the
class of c.e. and random reals in terms of halting probabilities of universal Chaitin machines, and we
will show that every c.e. and random real is the halting probability of some Solovay machine, that is,
a universal Chaitin machine for which ZFC (if sound) cannot determine more than its initial block
of 1 bits. A few open problems will be also discussed.