Answer by Asaf Karagila for fake and weak cardinals
This doesn't work. I'm leaving this as it might helpful to someone later on.I think that the answer is yes, but my idea has a gap.This happens for silly reasons: $V=L$ satisfies $\sf GCH$, and fake...
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Suppose $\lambda$ is a successor of a singular cardinal. We will say $\lambda$ fake if there is a transitive set $M$ such that $\lambda \subseteq M$ satisfying $\mathrm{ZFC}^-$ (ZFC without powerset)...
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