Is there a known result to the effect that it cannot be the case that for some natural $n$, there is a formula of length $n$ such that all cardinals can be defined by a formula whose length is shorter than $n$?

I'm speaking in the milieu of some fragment of a standard set theory extending first order logic with identity and membership, for example $\text{ZF-Regularity}$.

I've always expected that as regards the $\aleph_\alpha$ numbers the bigger it is the longer is the shortest definition of it. Or in general there cannot be a finite bound on definability of all of them. Is that correct?

Should the above be un-attainable, then is there a known argument against for example $n=10$ being such an upper bound?

Related is the following question: is it the case that definability of being * strictly subnumerous to $\aleph_\alpha$* is always equal or greater than definability of being

*for all $ \beta >\alpha$? By 'definability' I mean the shortest defining expression of course.*

**strictly subnumerous to $\aleph_\beta$*** After-note:* that Joel David Hamkins mentioned that the above linear assumption is FALSE, then my last question is:

is it the case that the definition of $< \aleph_i$ is always shorter than or equal in length to the defintion of $< \aleph_j$ as long as $i,j$ are both finite naturals and $i<j$?

finitecardinal number $k=2^{O(n)}$ (necessarily definable) that is not definable by a formula of length at most $n$. $\endgroup$ – Emil Jeřábek May 11 '18 at 10:242more comments