WHAT IS. . . a Halting Probability?
Turing’s famous 1936 paper “On computable numbers, with an application to the Entscheidungsproblem” deﬁnes a computable real number and uses Cantor’s diagonal argument to exhibit an uncomputable real. Roughly speaking, a computable real is one that one can calculate digit by digit, that there is an algorithm for approximating as closely as one may wish. All the reals one normally encounters in analysis are computable, like π, √2 and e. But they are much scarcer than the uncomputable reals because, as Turing points out, the computable reals are countable, whilst the uncomputable reals have the power of the continuum. Furthermore, any countable set of reals has measure zero, so the computable reals have measure zero. In other words, if one picks a real at random in the unit interval with uniform probability distribution, the probability of obtaining an uncomputable real is unity. One may obtain a computable real, but that is in- ﬁnitely improbable. But how about individual examples of uncomputable reals? We will show two: H and the halting probability Ω, both contained in the unit interval. Their construction was anticipated in..
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