# Is consistency strength order dense?

Definition: Let $\sigma$, $\theta$ be two statement in the language of set theory. We say $\sigma <_{c} \theta$ ($\sigma$ is strictly lower than $\theta$ in consistency strength order) if within ZFC, $Con(ZFC+\theta)$ implies $Con(ZFC+\sigma)$ but $Con(ZFC+\sigma)$ doesn't imply $Con(ZFC+\theta)$.

Question: Let $\varphi(x),\psi(x)$ are two formula in the language of set theory which define two notions of large cardinals in the following form:

$\varphi(x): x$ is an inaccessible cardinal with (first order expressible) property P.

$\psi(x): x$ is an inaccessible cardinal with (first order expressible) property Q.

If $\exists x \varphi(x)<_{c}\exists x \psi(x)$, is there a formula $\gamma (x)$ in the language of set theory in the following form

$\gamma(x): x$ is an inaccessible cardinal with (first order expressible) property R.

such that $\exists x \varphi(x)<_{c} \exists x\gamma(x) <_{c}\exists x \psi(x)$?

In the other words, is there a large cardinal notion strictly weaker than large cardinal notion defined by $\psi$ and strictly stronger than large cardinal notion defined by $\varphi$?