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Suppose $$j:M\prec N$$ is a non-trivial elementary embedding. Under what conditions on the sets (classes?) $M$ and $N$ (or even the critical point of $j$) does $j$ extend to an elementary embedding $$k:L(M)\prec L(N)?$$

In the case where $M=N=V_{\lambda+1}$, the existence of such a $k$ is a strictly stronger assumption (this is Woodin's $I_0$ axiom), but need this always be the case?

A more specific question involves extendible cardinals: Recall that $\kappa$ is extendible if, for every $\eta >\kappa$ there exists a $\theta>\eta$ and an elementary embedding $j:V_{\eta+1}\prec V_{\theta+1}$ such that $crit(j)=\kappa$ and $j(\kappa)>\eta$. Does $j$ extend to a $$k:L(V_{\eta+1})\prec L(V_{\theta+1})?$$

Is the existence of such a $k$ a straight-forward construction or is it strictly stronger than the existence of an extendible cardinal?

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Not sure if it helps - I was looking at: Slide 9 mentions that the new sets (of largest cardinality, those that are added only on the last level) must contain a closed set in a splitting tree. – Eran Jan 13 '13 at 20:46
Eran, I'm not sure that your comment has much bearing here. But maybe it does (I really don't know). The context of the slides you mention involves a strong limit cardinal $\lambda$ with countable cofinality. Given that particular context, one can develop analogues of descriptive set theory with tree representations of simple relations ($\Sigma_1^1$ and $\Sigma^1_2$ over $V_\lambda$). In fact, I think many of these analogues will hold even without the embedding assumption. – Everett Piper Nov 1 '13 at 2:02
Also, one of the relevant issues for strength may be (I think) that the full of axiom of choice is not available in the $I_0$ context. Perhaps if there is an issue with the presence of certain functions called $\omega$-J\'{o}nnson functions, then such an extension \emph{is} strictly stronger. An issue also could concern the behavior of partitions of certain stationary sets interacting with the embedding. I simply don't know at this point. My speculation is based on the now several different proofs of Kunen's Inconsistency. If you're interested, see Kanamori's book, chapter 23. – Everett Piper Nov 1 '13 at 2:08
up vote 3 down vote accepted

Not all such embeddings extend, as was mentioned at Bob Lubarsky's question on Extending elementary embeddings from initial segments to all of $V$.

Specifically, suppose that $\kappa$ is the least $1$-extendible cardinal. So there is an elementary embedding $j:V_{\kappa+1}\to V_{\eta+1}$. I claim that this embedding cannot extend to $L(V_{\kappa+1})\to L(V_{\eta+1})$, since the $j\upharpoonright V_{\kappa+1}$ is determined by $j''V_{\kappa+1}$, which is a size $2^\kappa$ subset of $V_{\eta+1}$, and by means of a flat pairing function all such subsets are coded as elements of $V_{\eta+1}$. Thus, $L(V_{\eta+1})$ would see that $\kappa$ is $1$-extendible, and so by elementarity there must be a $1$-extendible cardinal in $L(V_{\kappa+1})$ below $\kappa$, contradicting the minimality of $\kappa$.

A similar argument applies to the least $\theta+1$-extendible cardinal for any $\theta$.

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Joel, thanks! Extendible cardinals also have an upward reflective property, i.e. extendible cardinals imply the existence of certain (weaker) large cardinals above them (though as far as I understand, this does not go as far as a supercompact). Do you think you can give a crude estimate of how much stronger the following assumption is: some non-trivial elementary embedding $j$ witnessing a (partially) extendible cardinal extends to a non-trivial elem embedding $k$ with domain the relevant relatively constructible class $L(V_?)$. – Everett Piper Nov 1 '13 at 2:25
Everett, note that if you consider embeddings of the form $j:V_\eta\to V_\theta$, where $\eta$ is a limit ordinal, then the extenderr argument I gave on Bob Lubarsky's question shows that the embedding does lift to $j:L(V_\eta)\to L(V_\theta)$. – Joel David Hamkins Nov 3 '13 at 1:31

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