Cardinalities larger than the continuum in areas besides set theory As for my previous comment, a reference from the result about how $n$-hugeness suffices in the place of I3 is stated at least in Dougherty's paper "Critical points in an algebra of elementary embeddings."

Can one force there to be an elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$ for some inaccessible $\lambda$? Everett Piper. I am asking for an inaccessible cardinal $\lambda$ so that after forcing $\lambda$ becomes a singular strong limit cardinal of cofinality $\aleph_{0}$ and in the generic extension there is an elementary embedding from $V_{\lambda}$ to $V_{\lambda}$ which may be possible since $\lambda$ is after forcing a strong limit cardinal of countable cofinality. Are the questions that I asked clearly stated or should I reword them?