My apologizes if this question is not MO-appropriate.
Often I find myself wanting to state a hypothesis of a consistency result that can hold for an “aribtrary” regular cardinal. Only, it’s not really arbitrary, but it can be any regular cardinal below some large cardinal involved in the hypothesis. I’d like to state the result succinctly in a form like, “If ZFC is consistent with a measurable cardinal, then for every accessible regular cardinal $mu$, it is consistent that $varphi(mu)$.” But this doesn’t really make sense because the accessible cardinals are not terms in the formal language. I can revert to talking about $aleph_n$ for finite $n$, or $aleph_alpha$ for $alpha$ a recursive countable successor ordinal, but this seems to lose the generality. If I try talking about “definable cardinals,” then I run into the problem of non-absolute definitions.
Does anyone have any preferred elegant/eloquent general ways to address this? If possible, I’d prefer to talk about consistency results rather than describe the cardinals in terms of the relation between some models.