Let $\mathbf{L}$ be the set of all languages over $\Sigma=\{0,1\}$ and $E(\Sigma)$ be a lexicographic enumeration of $\Sigma^*$. Then there exists a bijection from $f:\mathbf{L} \rightarrow [0,1)$ where $f(L)$ has its $i$-th bit (to the right of the binary point) set to 1 iff the $i$-th string in $E(\Sigma)$ belongs to $L$. For simplicity, ignore languages that have a binary expansion consisting entirely of ones after a finite number of positions (such as .001001111111.....).
Questions
(1) Conjecture: If $L_{u}$ is an unrecognizable language, $f(L_u)$ must be an irrational , possibly transcendental number.
The intuition is that rational expansions are finite or periodic, hence must be trivially decidable, and algebraic numbers are expanded by solving the corresponding polynomial equation. Is there a proof/counterexample for this? Admittedly, this is only an intuition.
(2) What kind of numbers do the undecidable-but-recognizable languages map to?