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Suppose, in V, that $\kappa$ is a large cardinal, say $\kappa$ is supercompact, and $j:V \to M$ is an ultrapower embedding witnessing the large cardinal properties of $\kappa$. From here, we can take the seed hull of some element of M and get both of the following embeddings: a map $k$ which is the inverse collapse of the seed hull (let us call the transitive collapse the seed hull $M_0$), which, as it turns out, is an elementary embedding from $M_0$ to $M$, and a map $j_0: V \to M_0$. The map $j_0$ is an ultrapower map and we also get $j = k\circ j_0$.

If there is also a map $j_1: M \to M_0$ (maybe keep calling it $M_0$ even though it is contained inside or maybe contains $M_0$), an ultrapower embedding witnessing that $\kappa^M$ is supercompact, for example, then we could continue with the above construction. That is, we take a seed hull, get its transitive closure $M_1$, and then get $k_1$ an elementary map between $M_1$ and $M_0$ and an ultrapower embedding $j_{00}: M \to M_1$, with $j_1 = k_1 \circ j_{00}$.

It is possible that there is no map $j_1$ as above. In which case the construction is terminated. But, it could be that $\kappa$ is supercompact, or whatever $\kappa$ was in $V$, in $M$ as well. Potentially, how long could this construction go? Can it go forever, or must it necessarily stop at some infinite stage?

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I'm a little unclear about your final paragraph. If $\kappa$ remains supercompact in $M$, doesn't it follow that $M$ contains enough normal, fine ultrafilters to witness that $\kappa$ is supercompact? And if so, can't you simply take, in M, any one of these ultrafilters to form the ultrapower embedding $j_1$? – Everett Piper Oct 14 '12 at 21:42
Also, I am far from any sort of expert in this area, but your question, at least for $\kappa$ supercompact, seems to me to be related to the notions of enhanced supercompact, hypercompact, and weakly hypercompact cardinals. I believe they were introduced by Apter and Sargsyan while exploring some indestructibility-types of arguments. – Everett Piper Oct 14 '12 at 21:46
Hello, Erin. Can you give an example where $j_1$ exists? It seems a strange situation, since we would have $k:M_0\to M$ and $j_1:M\to M_0$. Or perhaps (I'm guessing) you meant a "new" $M_0$ here, not the same as what you originally called $M_0$? That is, where $\kappa$ remains supercompact after the supercompactness embedding, and you are doing it again? In this case, it is consistent that it happens as many (finite) times as you want, but the direct limit of $\omega$ many such embeddings will be ill-founded, as the threads starting with $\kappa$ will form a descending seqeuence in the limit. – Joel David Hamkins Oct 15 '12 at 1:51
Yep, I was thinking about some more, that maybe $j_1$ could be an embedding from $M$ to an ultrapower of $M$ contained in $M_0$, since it seems unlikely that this will be $M_0$. – Erin Carmody Oct 15 '12 at 13:30
Is it also consistent that it happens more than finitely many times? Doesn't it depend on $M$ and $V$ and $M_0$? Or, does it depend on the existence of such large cardinals? – Erin Carmody Oct 15 '12 at 19:14

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