## Recursively Enumerable Reals and Chaitin Ω Numbers

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Citations: | 34 - 3 self |

### BibTeX

@MISC{Calude_recursivelyenumerable,

author = {Cristian S. Calude and Peter H. Hertling and Bakhadyr Khoussainov and Yongge Wang},

title = { Recursively Enumerable Reals and Chaitin Ω Numbers},

year = {}

}

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### Abstract

A real is called recursively enumerable if it is the limit of a recursive, increasing, converging sequence of rationals. Following Solovay [23] and Chaitin [10] we say that an r.e. real dominates an r.e. real if from a good approximation of from below one can compute a good approximation of from below. We shall study this relation and characterize it in terms of relations between r.e. sets. Solovay's [23]-like numbers are the maximal r.e. real numbers with respect to this order. They are random r.e. real numbers. The halting probability ofa universal self-delimiting Turing machine (Chaitin's Ω number, [9]) is also a random r.e. real. Solovay showed that any Chaitin Ω number is-like. In this paper we show that the converse implication is true as well: any Ω-like real in the unit interval is the halting probability of a universal self-delimiting Turing machine.