Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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?

share|improve this question
    
Not sure if it helps - I was looking at: settheory.mathtalks.org/wp-content/uploads/2012/02/PR.pdf 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
add comment

1 Answer 1

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 $\theta$.

share|improve this answer
    
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
1  
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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

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