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Suppose that $\Phi(x,y)$ is a formula in the language of set theory so that for each natural number $n$, the axiom $\exists x\Phi(n,x)$ is a large cardinal axiom (for example consider $n$-huge cardinals). Then does $\text{Con}(ZFC+\exists x\Phi(n,x))$ for all $n$ imply $\text{Con}(ZFC+\forall n\exists x\Phi(n,x))$ (i.e. informally is the large cardinal heirarchy continuous in this sense or are there jumps in the large cardinal heirarchy)? Could there be a large cardinal axiom whose consistency strength is greater than each axiom $\exists x\Phi(n,x)$ for all natural numbers $n$ but whose consistency strength is strictly less than $\forall n\exists x\Phi(n,x)$?

Is it possible that there is a large cardinal axiom $\Psi$ so that $ZFC\models\text{Con}(\Psi)\leftrightarrow\forall n\in\omega\,\text{Con}(\Phi(n,x))$? What would the answer to these questions be if we replaced $\text{Con}(\Phi)$ with the existence of a transitive model that satisfies $\Phi$ or some other strengthening of mere consistency?

If one needs a formalization of what is meant by a large cardinal axiom, one can use this formalization due to Woodin mentioned in Mohammad Golshani's answer here or something similar to that.

I am also be interested in discontinuities in the large cardinal heirarchy by sequences of cofinality other than $\omega$ (for instance, for a sequence of cofinality $\aleph_{1},\mathfrak{c}$ or the first inaccessible cardinal), but I do not know of a nice way to formalize this notion.

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By compactness, if Con(ZFC$+\forall n\exists x\Phi(n,x)$) were a consequence of the infinite set of statements Con(ZFC$+\exists x\Phi(n,x)$), where $n$ ranges over genuine natural numbers, then it would already be a consequence of finitely many of those statements, and that's clearly not the case in non-trivial examples (like $n$-huge or $n$-inaccessible, provided of course that these are consistent).

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    $\begingroup$ And there are several well-known examples illustrating that the difference is witnessed in nature. The work of Harvey Friedman contains examples at the level of Mahlo cardinals, for instance. But the best known examples come from determinacy: $\mathsf{PD}$, projective determinacy, corresponds to the existence of (sharps for) inner models with $n$ Woodin cardinals, for all $n$. But an inner model with infinitely many Woodin cardinals corresponds to full $\mathsf{AD}$. $\endgroup$ – Andrés E. Caicedo Jul 23 '15 at 19:03
  • $\begingroup$ Andreas Blass. It seems like I completely overlooked the compactness theorem when posting this question. I was afraid of something like that. $\endgroup$ – Joseph Van Name Jul 28 '15 at 1:12
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    $\begingroup$ Andres Caicedo. Perhaps you should give an answer elaborating on your comment. Since your comment has already been highly upvoted, I am sure that the people on MO with an affinity towards set theory would appreciate your answer. $\endgroup$ – Joseph Van Name Jul 28 '15 at 1:13

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