Nice way of referring to “some cardinal that always exists”

Question

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.

EDIT: Does the following notion make sense? Suppose $\varphi(x)$ is a formula in the language of set theory. We say $\kappa_\varphi$ is an absolute cardinal if ZFC proves that there is a unique cardinal $\kappa$ such that $\varphi(\kappa)$, denoted $\kappa_\varphi$, and for any two models $M,N$ of ZFC with the cardinals of one being an initial segment of the cardinals of the other, $\kappa_\varphi^M =\kappa_\varphi^N$. Does this notion nicely capture examples of consistency results like I mentioned? For instance, if ZFC+inaccessible is consistent, then for any absolute successor cardinal $\kappa_\varphi$, it is consistent that there are no Kurepa trees on $\kappa_\varphi$. Are there examples that make this a bad definition?

Answer 1

Unless I'm misreading, there's an implicit identification in the question of "cardinal that always exists" with "cardinal below the first [large cardinal]." I don't really buy such an identification; the answer below treats the second notion, that is, the question of whether we can talk about "each cardinal below the first [large cardinal]" in a model-free way. This may not be relevant to the OP after all, in which case I'll delete this.

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I don't think there is one, at least not without a lot of circumlocution - and I don't think there should be.

To me there's a pretty fundamental ambiguity when one says anything like

Assuming Con($\mathsf{ZFC}$ + a measurable), for every uncountable regular cardinal $\kappa$ below the first measurable it is consistent that $\kappa$ has property $P$

in the first place. Since in general there's no way to refer to a single arbitrary cardinal in the language of set theory alone, I'm not sure how to make sense of consistency in this context. My best interpretation of "consistent" in this context is in the sense of outer models, so that I'd interpret the above as saying:

Suppose $M$ is a model of $\mathsf{ZFC}$ + a measurable. Let $X$ be the set of uncountable regular cardinals below the least measurable, in the sense of $M$. Then there is a family $(M_\kappa)_{\kappa\in X}$ such that each $M_\kappa$ is an outer model of $M$ satisfying "$\kappa$ has property $P$."

However, this may not be what's intended - e.g. maybe we should restrict attention to countable $M$, or well-founded $M$, or both. Conversely, maybe we want to say more about those outer models - maybe each $M_\kappa$ is a set forcing extension of $M$ (even "uniformly" in an appropriate sense), or maybe each $M_\kappa$ still satisfies "$\kappa$ is a regular cardinal below the least measurable."

So the use of proof-theoretic language here really seems to obscure the actual meaning of the statement in question, to the point that genuine information is lost. Honestly I prefer it when authors don't use this sort of phrasing at all (or if they do, I prefer it if they give their notion of "consistency" a very precise definition somewhere else in the paper).