Virtually extendible cardinal
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Virtually extendible cardinal is the virtual version of an extendible cardinal. That is, the defining elementary embedding is not in V itself, but rather in a forcing extension of V. This weakens the cardinal substantially and makes it consistent with V=L.
Strength relative to other large cardinals
virtually extendible cardinal is stronger than 1-iterable cardinal and remarkable cardinal, but weaker than 2-iterable cardinal.
References
- Gitman, Victoria; Schindler, Ralf (December 2018). "Virtual large cardinals". Annals of Pure and Applied Logic. 169 (12): 1317–1334. doi:10.1016/j.apal.2018.08.005.
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