User blog comment:Fejfo/bitstring lexographical notation/@comment-35470197-20190913005559

Good! If you allow the lack of an effective algorithm, then the formalisation can be done if you fix an ordinal \(\alpha\) equipped with a system of fundamental sequence such that for any limit ordinals \(\beta\) and \(\gamma\) below \(\alpha\), \(\beta[2] = \gamma[2]\) implies \(\beta = \gamma\). Indeed, you can define an injective map \begin{eqnarray*} \textrm{encode} \colon \alpha & \to & \bigcup_{n=1}^{\infty} \{0,1\}^n \beta & \mapsto & \textrm{encode}(\beta) \end{eqnarray*} in the following formal way: So the main issue is to find a system of fundamental sequences satisfying the desired injectivity. I think that the standard fundamental sequences for Wainer hierarchy satisfies it. Therefore it can be formalised up to \(\varepsilon_0\) as you desire if I am correct. I do not know whether the same is true for the standard fundamental sequences for Venlen hierarchy.
 * 1) If \(\beta = 0\), then \(\textrm{encode}(\beta) := 0\).
 * 2) If \(\beta\) is a successor ordinal, then \(\textrm{encode}(\beta) := \textrm{encode}(\beta[0])0\).
 * 3) If \(\beta\) is a non-zero limit ordinal, then \(\textrm{encode}(\beta) := \textrm{encode}(\beta[2])1\).