Is there still any hope that the GCH could be equivalent to some large cardinal axiom?

Question

Is there still any hope that the GCH could be equivalent to some large cardinal axiom?

Even a simple yes or not answer will be fine. Thanks!!

Answer 1

No.

The term "large cardinals" has no explicit and well-defined meaning, but it's general meaning is an additional axiom which is strictly stronger in consistency than $\sf ZFC$, that is some $\varphi$ such that $\sf ZFC+\varphi$ can prove the consistency of $\sf ZFC$.

On the other hand, $\sf ZFC+GCH$ is equiconsistent with $\sf ZFC$ so it is not a large cardinal axiom per se.

(To slightly elaborate on the above meaning of no well-defined meaning, $0^\#$ is considered a large cardinal axiom, but it really just says that a certain real number exists; or a Jonsson cardinal which is not even an inaccessible one, but its existence implies that inaccessible cardinals do exist. Both these, and more, are considered large cardinal axioms despite not being directly related to inaccessible cardinals and so.)

Answer 2

I don't think there was ever any hope that the GCH would be equivalent to a large cardinal axiom. There was once a hope that CH or its negation would be implied by a large cardinal axiom, but this was dashed by the Levy–Solovay theorem and its generalizations, which say that large cardinals are preserved by "small" forcings such as the usual ones to force CH and not CH.