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(I ran into this question while thinking about the (Strong) Inner Model Hypothesis: see http://www.jstor.org/stable/4093051 or http://arxiv.org/abs/0711.0680.)

A measurable cardinal is a cardinal $\kappa$ such that there is an elementary embedding $j: V\rightarrow M\subseteq V$ with $M$ an inner model of $V$ and $crit(j):=\min\lbrace \alpha\in ON: j(\alpha)\not=\alpha\rbrace=\kappa$.

Now we can allow the target model of $j$ to live, not inside $V$, but inside some set-generic extension of $V$ as follows. Say that $\kappa$ is outer-measurable if there is some poset $\mathbb{P}\in V$, some $G$ which is $\mathbb{P}$-generic over $V$, and some transitive inner model $M$ of $V[G]$ such that there is an elementary embedding $$ j: V\rightarrow M\subseteq V[G]$$ with $crit(j)=\kappa$.

In general, given any large cardinal property $(\*)$ defined in terms of elementary embeddings, we can define outer-$(\*)$-ness to be the property $(\*)$ where the target model $M$ is allowed to be an inner model of some set-forcing extension of $V$, rather than $V$ itself. My questions, then, are:

  1. Is there a large cardinal property $(\*)$ such that we can have an outer-$(\*)$ cardinal which is not $(\*)$?

  2. Is there a large cardinal property $(\*)$ such that the consistency strength of an outer-$(\*)$ cardinal is weaker than the consistency strength of a $(\*)$-cardinal?

I suspect that the answer to the second question is "no;" I have no idea about the first question.

[EDIT: Thanks to Joel for pointing out that my now-removed claim that "measurable=outer-measurable" is wrong.]

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If there is a precipitous ideal $I$ on $\omega_1$---a hypothesis equiconsistent over ZFC with the existence of a measurable cardinal---then after forcing with $P(\omega_1)/I$, we get an elementary embedding $j:V\to M\subset V[G]$ with critical point $\omega_1$. Thus, on this hypothesis, $\omega_1$ is outer measurable in the sense you have described. This answers question 1 and also shows that your assertion that

ZFC proves that a cardinal is outer-measurable only if it is measurable.

is not correct if these large cardinal hypotheses are consistent. Similar phenomenon arise with analogues of many of the other large cardinal concepts, such as supercompactness and others. Matt Foreman has particularly emphasized the richness of these generic large cardinal concepts, and he has explored this topic extensively. Some of this material is in his chapter in the Handbook of Set Theory.

Perhaps you were thinking of the related assertion, which I believe is correct (but this should be confirmed by inner-model theorists), namely, if $\kappa$ is outer measurable, then there is an inner model where $\kappa$ is measurable. That is, if $\kappa$ is the critical point of a generic embedding, then I believe one can still construct the canonical inner model $L[\mu]$ in which $\kappa$ is measurable.

Finally, you seem to indicate that you believe that the concept of outer-measurability is not first-order expressible in ZFC. But I think it is expressible: $\kappa$ is outer measurable if and only if there is a partial order $\mathbb{P}$ forcing that there is a $V$-ultrafilter $\mu$ on $\kappa$ such that $\text{Ult}(V,\mu)$ is well-founded.

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You're right, I didn't think my argument through for measurability. Re: your last paragraph, I just wanted to acknowledge that things as I'd written them were not trivially translatable into ZFC, in case that came up. –  Noah S Mar 10 '13 at 3:03
    
Also, for others who like me don't know much about precipitous ideals, this looks like a good introduction: "Precipitous ideals" by Jech, Magidor, Mitchell and Prikry (jstor.org/stable/2273349?seq=1). –  Noah S Mar 10 '13 at 3:07
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Looking at Foreman's article ("Ideals and Generic Elementary Embeddings," link.springer.com/chapter/…), I think it's addressing my question exactly, but I'm having a bit of trouble understanding it. A specific example: the hypothesis of Theorem 5.1 on page 959 seems to be not fully stated. What role does the generic $G$ play in the hypothesis? My guess is that $j$ and $M$ are classes in $V[G]$, but I just want to check. –  Noah S Mar 10 '13 at 7:16
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