Turing’s Halting theorem

Are there any noncomputable numbers They are certain to exist since the cardinality of the set of all possible programs (which is equal to that of the integers) is less than the cardinality of the set of reals. It is, in fact, easy to construct a noncomputable number, using Turing s Halting Theorem number, in some arbitrary manner, all possible programs that run with, say, a single input instruction Pi (-> l,p2 2,...,pn n,..., and set ( = n2, where q = 1 if the... 

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