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?


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 $V_\kappa\prec V_\lambda$.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.