# consistency of large cardinal axiom

It is known that if ZFC is consistent, ZFC+"no such large cardinal exists" is consistent. Then, is ZFC+"such large cardinal exists" known to be consistent? This would imply that proving large cardinal axiom is independent from ZFC...

Thanks.

-

If, however, $\kappa$ is an inaccessible cardinal then $V_\kappa$ which is the collection of all sets whose von Neumann rank is $<\kappa$ is a model of ZFC. Due to the completeness theorem of Godel, this implies that ZFC is consistent.
Furthermore, suppose ZFC+There is an inaccessible cardinal is consistent, let $\kappa$ be the least inaccessible cardinal, then $V_\kappa$ is a model of ZFC but there are no inaccessible in $V_\kappa$.
(and almost all large cardinals are inaccessible, and if they are not inaccessible they imply that below them there are inaccessible cardinals, it might be that a cardinal is weakly inaccessible but "going down" to $L$ makes it a strongly inaccessible, and we are only interested in consistency results anyway)