Is there a set $S$ that is definable in ZFC and known to be infinite, but for which we know neither the aleph nor beth number?
For example, we do not know the aleph number of $|\mathbb R|$, but we know its beth number, $|\mathbb R| = \beth_1$. Conversely, we trivially know the aleph number of $\aleph_1$, but we do not know its beth number (or whether it has one).
To be clear, by "not knowing the beth number of $S$" I mean we either know $S$ has a beth number but don't know the particular number, or we don't even know whether $S$ has a beth number. Also, by "not knowing" $X$ I mean either we have shown $X$ to be undecidable in ZFC, or we just don't know in the usual sense.