Turing and Lebesgue enter Cantor's Cathedral: Do the computable reals form a set of measure zero?
No one shall expel us from the paradise which Cantor has created for us.Hilbert
The computable reals in \([0,1]\) are countable and therefore we may denote them by the set \(R = \{r_i\}_{i=1}^{\infty} \subset [0,1]\). We’ll note that we can cover \(r_1\) with \(\frac{\epsilon}{2} > 0\), \(r_2\) with \(\frac{\epsilon}{4} > 0\), and \(r_n\) with \(\frac{\epsilon}{2^n} > 0\) so the total length of the cover is [1]:
\begin{equation} \sum_{n=1}^\infty \frac{\epsilon}{2^n} = \epsilon \end{equation}
and since \(\epsilon\) may be made arbitrarily small, the computable reals in \([0,1]\) form a set of measure zero.
The noncomputable reals, on the other hand, may be constructed using an infinite sequence of coin tosses. So if we represent the noncomputable real number \(x = \{x_i\}_{i=1}^\infty\) in base2 so \(x \in \{0,1\}^\infty\), we have:
\begin{equation} x_i \sim \text{Ber}\big(\frac{1}{2}\big) \end{equation}
Now, here’s what I find both fascinating and mysterious. There is a large class of computable numbers, the irrational algebraic numbers, which are believed to be absolutely normal without exception. This means that their base2 expansion has statistical behaviour that is indistinguishable from a binary sequence generated by coin tosses and therefore they are not finitestate compressible.
This class of numbers, the irrational algebraic numbers \(\mathcal{A}\), is of great interest to AI researchers as it implies that given \(x \in \mathcal{A}\) no machine learning system designed using any technology may be used to reliably predict the Nth bit of \(x\) given the first \(N1\) bits of \(x\). And yet an exact formula exists for computing the Nth bit of \(x\).
References

R. Courant and H. Robbins (1941), What is Mathematics?, Oxford: Oxford University Press.

Gregory Chaitin. ALGORITHMIC INFORMATION THEORY. Cambridge University Press. 1987.

Godofredo Iommi. BESICOVITCH FORMULA. Thermodynamic formalism and applications. 2012.