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If excluded middle holds then the function is computable, as follows. Consider the set $$S = \{n \in \mathbb{N} \mid \text{there are $n$ consecutive 9's in $\pi$}\}.$$ By excluded middle $S$ is bounded or unbounded. Consider both possibilities: If $S$ is bounded then it has a largest element $m \in S$. The function $f$ can be computed as follows: on input $...

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