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Let $\kappa$ be measurable cardinal(or other large cardinals) and let $j:V\longrightarrow M$ witnessing this. We know that $j(\kappa)$ has large cardinal properties in $M$, but what about $j(\kappa)$ in $V$.

Let me give a couple tentative nonstandard definitions:

$\bullet$ Call a measurable cardinal $\kappa$ weakly compact preserving, if there is $j$ an elementary embedding witnessing measurability of $\kappa$, such that $j(\kappa)$ is weakly compact in $V$.

$\bullet$ Call a measurable cardinal $\kappa$ reflecting preserving if $V_\kappa\prec V_{j(\kappa)}$, again in $V$.

What I am really interested in is a weakly compact preserving-reflecting preserving measurable cardinal $\kappa$ such that there is a weakly compact cardinal $\lambda>\kappa$ such that $V_\lambda\prec V_{j(\lambda)}$.

Question: Where such a cardinal take place in large cardinals hierarchy?

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If $\kappa$ is measurable and there is a weakly compact cardinal $\lambda$ above, then there is an elementary embedding $j:V\to M$ with critical point $\kappa$ and $j(\kappa)=\lambda$. The reason is that one may simply iterate a normal measure, which pushes $j(\kappa)$ higher, until it hits that $\lambda$. The same argument works with any kind of large cardinal $\lambda$. But the embedding $j$ is not an ultrapower embedding. Of course, with an ultrapower embedding by an ultrafilter on $\kappa$, the value of $j(\kappa)$ is never a cardinal in $V$, since it is strictly between $2^\kappa$ and its successor.

For your second question, if $V_\kappa\prec V_\lambda$ for a class club of $\lambda$, then again we can find $j:V\to M$ with $j(\kappa)$ being one of those $\lambda$, by iterating. The consistency strength of that situation is strictly higher than a proper class of measurable cardinals, since $V_\kappa$ itself must be a model of that theory. But it is less than a stationary proper class of measurable cardinals, since from that assumption one can find a model with a measurable cardinal $\kappa$ with $V_\kappa\prec V$ and indeed a stationary class of measurable $\lambda$ above which form an elementary chain $V_\kappa\prec V_\lambda$.

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  • $\begingroup$ Thank you, it is nice, I need time to take a careful look at. $\endgroup$ – Rahman. M Mar 22 '17 at 11:27
  • $\begingroup$ What about the situation one wants to control cardinals above critical point. I mean $\kappa$ is measurable, $\lambda>\kappa$ is weakly compact and $V_\kappa\prec V_{j(\kappa)}$ and $V_\lambda\prec V_{j(\lambda)}$ such that $j(\lambda)$ is inaccessible or at least wordly. $\endgroup$ – Rahman. M Mar 24 '17 at 9:29
  • $\begingroup$ Note that if we relax the phrasing of the first question a bit, we can get positive results: Let $U$ be a 'sufficienlty nice' measure on $\kappa$ (e.g. be derived from an ultrapower embedding) and let $i_U \colon L \to L$ be the restriction of the ultrapower embedding to $L$. This itself is an ultrapower embedding of the structure $(L; \in, U \cap L)$ (i.e. definable in this structure by the usual construction) and in there $i_U(\kappa)$ is weakly compact. $\endgroup$ – Stefan Mesken Mar 30 '17 at 15:43

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