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Aug 4, 2011 at 20:15 comment added Ali Enayat @Andreas: your characterization can be fine-tuned by adding $Ult(V,U)=M$ precisely when every element of $M$ is definable in $M$ by a first order formula whose parameters are allowed to come from {$\kappa$} $\cup$ the range of $j$.
Aug 4, 2011 at 19:38 comment added Asaf Karagila Now that you mention that, I think that I may have seen this sort of example of why $M$ need not be equal to $\operatorname{Ult}(V,U)$. Many thanks for the answer!
Aug 4, 2011 at 19:33 history answered Andreas Blass CC BY-SA 3.0