Can You Compute This?

A computable number is one which, given enough time, can be calculated to whatever degree of precision is desired. For example, the irrational number which has a 1 in every prime slot \[ 0.01101010001010001 \ldots \] is computable. To calculate it to an accuracy of \( 10^{-n} \), we just need to determine which of the first \(n\) positive integers are prime.

Are there countably many or uncountably many computable numbers?

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