Does there exist a computable function $f:\mathbb{N}\rightarrow \mathbb{Q}$ such that:

  • For all $t\in\mathbb{N}: 0\le f(t) < X$
  • $\lim\limits_{t\rightarrow\infty} f(t) = X$

Where $X$ is an uncomputable real number.

The only reference to this question I have found was the answer to this question: https://math.stackexchange.com/a/1052579/168764, where the function seems that it would hold, but I have no idea how to prove that the limit of this function is an uncomputable real number.

  • $\begingroup$ I believe this answer I wrote three years ago answers your question: math.stackexchange.com/a/1267124/161559 $\endgroup$ – kasperd Apr 2 '18 at 23:34
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    $\begingroup$ The numbers obtainable as such as limit $X$ are called left-c.e. reals, in case you want to search for more about their properties. $\endgroup$ – Arno Apr 3 '18 at 9:11
  • $\begingroup$ maybe also math.stackexchange.com/a/462835/128985 which gives such a function I think (unless I have the logic the wrong way around) $\endgroup$ – Philip Oakley Apr 3 '18 at 17:06

Consider the real number encoding of the (almost) halting problem, i.e. $0.r_1r_2...$ where $r_i=1$ if the i'th Turing machine (relative to the lexicographic ordering) halts on the empty input, and $r_i=0$ otherwise. Let us denote this number by $R$.

Now, consider the machine $M$ which on input $n$ simulates all Turing machines of length $< n$ on the empty input for $n$ steps, and returns $0.\hat{r_1}...\hat{r_{2^n-1}}$ where $\hat{r_i}=1$ if the $i$'th Turing machine halts on the empty input in less than $n$ steps, and $\hat{r_i}=0$ otherwise. Clearly for all $n$ it holds that $M(n)< R$, and it is not too hard to show that $\{M(n)\}_{n\in\mathbb{N}}$ converges to $R$. The key point is that rate of convergence is not computable, meaning that given $\epsilon$, you cannot compute the index such that beyond it the series is $\epsilon$-close to $R$.

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  • $\begingroup$ The $\epsilon$ you mentioned is any real number or is it a computable real number? (Does it make a difference?) $\endgroup$ – Pedro A Apr 3 '18 at 20:25
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    $\begingroup$ There isn't any computability issue here, but since we're talking about an input to a Turing machine, it has to have some finite representation, so we can think of $\epsilon$ as a small rational number. $\endgroup$ – Ariel Apr 3 '18 at 20:57

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