# Let $\Phi$ denote the statement that $\mathrm{GCH}$ holds, and that no inaccessible cardinals exist. Is $\Phi$ limiting?

By an "inner model," let us mean a transitive subclass of the universe satisfying $\mathrm{ZFC}.$ Given that, here's an (intentionally) vague definition.

Definition. Call a set of axioms $\Phi$ in the language of set theory limiting iff there exists a (consistent, as far as we know) large cardinal axiom $\lambda$ such that no model of $\mathrm{ZFC}+\Phi$ has an inner model satisfying $\lambda$.

Example. $\{V=L\}$ is limiting, since (if I recall correctly) it contradicts the existence of $ω_1$-Erdős cardinals, and implies that the universe has no proper inner models.

Anyway, here's an (intentionally) vague question.

Question. Let $\Phi$ denote the statement that the generalized continuum hypothesis holds, and that no inaccessible cardinals exist. Is $\Phi$ limiting?

• This might be more of a question for MathOverflow? – Henno Brandsma Jun 1 '14 at 11:10
• If $\lambda$ isn't really strong (I think mostly everything below a huge cardinal should work) we can take a model of GCH and $\lambda$ with no inaccessibles above it and then collapse the large cardinal. – Miha Habič Jun 1 '14 at 19:36
• $\Phi$ is not limiting as far as we know. – Andrés E. Caicedo Jun 1 '14 at 22:43