This is a question that I always had in mind.
When it is said that the consistency of a theory $T$ requires the assumption of existence of some specified cardinal $\kappa$. Is that taken to mean that $T$ can prove the existence of all cardinals strictly smaller than $\kappa$?
A clarifying example, suppose its said that a theory $T$ requires existence of $0^\sharp$, one might get the impression that $T$ can interpret each theory ZFC+ $\kappa$, where $\kappa$ is a cardinal property such that ZFC + $\kappa$ is interpretable in ZFC+$0^\sharp$. So viewing the list of large cardinal properties, then one would for example expect that $T$ can interpret ZFC+ Mahlo cardinal exist. Is that impression correct?