@NoahSchweber: then "$|\beta|$" denotes "the cardinality (cardinal number) of the cardinal $\beta$"?

@NoahSchweber: Thank you for the explanation. Yes, of course, $\alpha^{+L} = \omega_1$ is only one of conditions, not a separate lemma. I need to clarify one thing: you said that $\beta$ is "the unique ordinal", then "the smallest cardinal". Did you mean that $\beta$ is "the smallest initial ordinal"? I am asking because the notions of cardinals and ordinals imply different things.

"I'm not sure what your objection is." — there was no objection; I just wanted to clarify a few things that are somewhat unrelated to the answer. "Which do you actually want to know about here?" — I wanted to know about the heights of the minimal rank of set models of uncomputably axiomatizable theories (such theories are mentioned in your comment on Math.SE). (Maybe this question needs a separate post.) Sorry for the possible confusion and thank you for the explanations!

"this has nothing to do with the details of T (beyond its computable axiomatizability)" — wait, but the question of this post implies that there is no such limitation. Pick any consistent theory T from the set of all infinite binary sequences (note that T may be uncomputably axiomatizable, and the complexity of axioms is unrestricted in the Lévy hierarchy). Then T must have a set model $S$. How large can the minimal rank of $S$ be? Will the answer depend on the definition of rank?

Then if $S$ is a set that models such T, what is the minimal rank of $S$?

I have read the answers to this question on Mathoverflow, and now I have another question: if T = ZFC + "there exists a supercompact cardinal" is consistent, does T have a set model?

@ArvidSamuelsson: thank you for the explanation. Yes, my statement "$\alpha$ must be much smaller than $\beta_{\mathcal{L}}$" should be "$\alpha$ (if it exists) must be much smaller than $\beta_{\mathcal{L}}$"...

It seems that I need an explanation for the claim that $\beta_{\mathcal{L}}$ is less than the least supercompact. Consider the theory T = ZFC + "there exists a supercompact cardinal", which corresponds to some infinite binary sequence (existence of a supercompact cardinal is $\Sigma_3$ in the Lévy hierarchy). What is the smallest ordinal $\alpha$ such that $V_{\alpha} \models \text{T}?$ Note that $\alpha$ must be much smaller than $\beta_{\mathcal{L}}.$

@KConrad: the point of $g(n)$ is some measure of how "rich" a number can be if we look at its representation in different bases. The point of $r(n)$ is give a reasonable upper bound for the bases.