X-Nico

Regular cardinal

Furthermore, it is consistent with ZF that every aleph bigger than $\backslash aleph\; 0$ is singular (a result proved by Moti Gitik).

Singular cardinals hypothesis

Gitik, building on work of Woodin, was able to replace the supercompact in Silver's proof with a measurable of Mitchell order $\backslash kappa^\{++\}$.

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In fact, by results of Moti Gitik, ZFC + the negation of SCH is equiconsistent with ZFC + the existence of a measurable cardinal κ of Mitchell order κ⁺⁺.

see also