See Eilenberg/Schutzenberger, On pseudovarieties for background on pseudovarieties. I've phrased things in terms of pairs-of-sets to avoid some annoying language about multisets. Also, I'm aware that Reiterman's theorem is in many ways a better characterization of pseudovariety-hood, but it doesn't give rise to the sort of question I'm asking here.
Suppose $\mathscr{V}$ is a pseudovariety in a countable language $\Sigma$. Say that a minimal description of $\mathscr{V}$ is a pair $(E,F)$ of sets of $\Sigma$-equations such that
$\mathscr{V}$ is the set of finite $\Sigma$-algebras which satisfy all equations in $E$ and all-but-finitely-many equations in $F$; and
if $E'\subseteq E, F'\subseteq F$ are such that $(E', F')$ satisfies the previous bulletpoint, then $E'=E$ and $\vert F\setminus F'\vert<\infty$.
Two minimal descriptions $(E,F), (G,H)$ are equivalent iff $E=G$ and $F\Delta H$ is finite (where "$\Delta$" denotes the symmetric difference).
What are the possible numbers of minimal descriptions (up to equivalence) of a pseudovariety?
For example, is there a pseudovariety with exactly 3 minimal descriptions up to equivalence? More set-theoretically, is it consistent with $\mathsf{ZFC+\neg CH}$ for there to be a pseudovariety with exactly $\aleph_1$ minimal descriptions up to equivalence?
(Re: that last question, unless I'm missing something the set $M$ of minimal descriptions of $\mathscr{V}$ is coanalytic and equivalence on that set is Borel, so by Burgess' trichotomy theorem we can't get anything strictly between $\aleph_1$ and $2^{\aleph_0}$. Specifically, view $M$ as a coanalytic subset of an appropriate Polish space $\mathcal{X}$ and consider the equivalence relation which identifies two elements of $M$ iff they are equivalent in the above sense and which collapses the whole complement of $M$ to a single class. This is analytic, so Burgess applies. I don't see how to make it coanalytic though, so Silver doesn't seem to help here.)
