A thought on the computability of numbers and the continuum hypothesis

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10.08.2020

So last night I wondered whether all real numbers were computable.
Because if that was the case, then every real number could be matched to a turing machine (the one that computes it), which in turn could be matched to a natural number (its Gödel numbering). But this would imply \(\aleph_{0} = \aleph_{1}\), which is a contradiction, because the reals are isomorphic to the powerset of the natural numbers. So there are reals that are not computable.

Is this the case because computations have to be deterministic and finite? But you can calculate \(\pi\)? Or are approximations invalid? Because calculating \(\pi\) exactly is not finite, is it?

Okay, so I just took a look and I’m really not surprised that they just brought up the Halting Problem again.


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