If $\kappa$ is a strongly compact cardinal, then the singular cardinal hypothesis holds above $\kappa$. Hence the existence of large cardinals at the level of "strongly compact" or above is incompatible with even (apparently) mild large powerset axioms like "$2^\kappa$ always strictly exceeds $\kappa^+.$"

This raises the question:

Is there a "large powerset axiom" $\varphi$ so extreme that $\mathrm{ZFC}+\varphi$ disproves the existence of strongly inaccessible cardinals? Let us also require that $\mathrm{ZFC}+\varphi$ does not prove $\neg \mathrm{Con}(ZFC + \varphi).$

petitio principii", itself a poor medieval translation from Greek to Latin, as argued by those pesky "descriptivist" linguists here: languagelog.ldc.upenn.edu/nll/?p=2290 Used correctly (to mean "assume the conclusion"), chances are it will be misunderstood. Used incorrectly around certain sophisticated people, it becomes a shibboleth. Best to avoid it altogether. $\endgroup$skunked: "sticking to the older sense confuses those unfamiliar with it, while using the newer sense annoys traditionalists who feel that it is wrong." $\endgroup$2more comments