Acceptable numberings are computably intertranslatable

By [L1], let $({\phi }_e)$ denote the standard acceptable numbering coming from the fixed machine coding. Because $\alpha$ is universal by [L2], the partial function $(e,x)\mapsto {\phi }_e(x)$ has some $\alpha$-index $u$. Because $\alpha$ is acceptable, [L2] supplies a total computable hard-wiring map $s_{\alpha }$, and then $t_{\phi \to \alpha }(e):=s_{\alpha }(u,e)$ satisfies ${\alpha }_{t_{\phi \to \alpha }(e)}(x)\simeq {\alpha }_u(<mi\; mathvariant="normal">⌜</mi>e,x{<mi\; mathvariant="normal">⌝</mi>}_{\mathrm{s}\mathrm{e}\mathrm{q}})={\phi }_e(x)\mathrm{.}$ So $\phi$ translates computably into $\alpha$. The same argument gives a computable translator $t_{\phi \to \beta }$ from $\phi$ into $\beta$.

Because $\phi$ is universal and $\alpha$ is acceptable, the same argument with the roles reversed gives a total computable translator $t_{\alpha \to \phi }$ with ${\phi }_{t_{\alpha \to \phi }(e)}={\alpha }_e\mathrm{.}$ Likewise there is $t_{\beta \to \phi }$ with ${\phi }_{t_{\beta \to \phi }(e)}={\beta }_e\mathrm{.}$

Compose the translators through the hub numbering $\phi$: $t_{\alpha \to \beta }:=t_{\phi \to \beta }\circ t_{\alpha \to \phi },t_{\beta \to \alpha }:=t_{\phi \to \alpha }\circ t_{\beta \to \phi }\mathrm{.}$ Then ${\beta }_{t_{\alpha \to \beta }(e)}={\phi }_{t_{\alpha \to \phi }(e)}={\alpha }_e,$ and similarly ${\alpha }_{t_{\beta \to \alpha }(e)}={\beta }_e$.

Therefore any two acceptable numberings are computably intertranslatable.