Everett Piper
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 Nov 9 revised Can one force there to be an elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$ for some inaccessible $\lambda$? argument below the theorem is not a proof of the theorem, just an example of some subtleties of the theorem, and the example (I hope) partially answers the question. Nov 9 answered Can one force there to be an elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$ for some inaccessible $\lambda$? Nov 8 comment Can one force there to be an elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$ for some inaccessible $\lambda$? extension. These facts are all true at least in the case when a small forcing is used. There are large forcings which will destroy certain large cardinals, large forcings which will create so-called generic large cardinals, and, if I remember correctly, forcings that will resurrect large cardinals which were previously killed. There are many articles exhibiting this type of phenomenon, though I don't have any specific references at the moment. Nov 8 comment Can one force there to be an elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$ for some inaccessible $\lambda$? @JosephVanName. Your question is vague (to me at least) but now I think I get what you are after. The point of my first comment was that the consistency of an I0 gives an I1 with your embedding in a generic extension, similarly I1 suffices for an I3 with that property. These embedding already exist in the ground model and are preserved to the generic extension. This is one direction of the Levy-Solovay phenomenon. The other direction is that large cardinals are not created in the generic extension. So your desired $\lambda$, if it is not already I3, say, then it will not become I3 in the Nov 8 comment Can one force there to be an elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$ for some inaccessible $\lambda$? Maybe I've misunderstood your question. Are you asking for an inaccessible above an I3 which is itself not I3, that you can collapse and find a non-trivial embedding in the generic extension? Or do you want an inaccessible (or perhaps something stronger), not necessarily above an I3 that you can singularize through forcing and introduce an embedding in the generic extension? Or maybe something else altogether? Nov 8 comment Can one force there to be an elementary embedding $j:V_{\lambda}\rightarrow V_{\lambda}$ for some inaccessible $\lambda$? @JosephVanName. Don't I1 cardinals in V already imply the existence of many I3 cardinals below? If we use a small partial order, these I3 cardinals are preserved to the generic extension. More specifically, if $\lambda$ is I1, then there is a $\bar{\lambda}$ below it and a $j$ a witnessing that it is I3. If $G\subset P$ is generic and \$|P|