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The existence of not computable real numbers has been eating at me for a long time.

However, I have not seen any example of such a number not related to the halting problem. Could any such number be defined?

Below are my thoughts which are not part of the question and can be ignored:

I can understand intuitively why NC numbers appear in the HP, because we are basically asking a certain class of machines to 'compute themselves'. Or rather, we are searching for a machine which knows everything about all the other possible machines. Not likely to happen.

However, couldn't the problem be solved by generalizing the definition of computable functions? For example, taking quantum computing into account?

Yuriy S
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  • Related: http://math.stackexchange.com/questions/462790/are-there-any-examples-of-non-computable-real-numbers – symplectomorphic Dec 26 '16 at 20:47
  • Note that the reals computable by quantum computers are exactly those computable by regular computers - the only difference is the difficulty of computing them (in the complexity-theoretic sense). A better physical attempt to get beyond Turing computability is via black holes, but it's rough on the computer (and relies on a simplistic model of black holes). – Noah Schweber Jan 01 '17 at 22:20

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