elementary embeddings $j$ in set theory with $V$ and $M$

Question

I'm confused by a variety of elementary non-trivial elementary embedings $j$ we might have. There are 9 "syntactical" possiblities;Here $M$ is a transitive model. I'll name them with a wish that someone would tell me which are impossible, which are not useful and which are strong cardinals-like: $$j_1:L\to L,$$ $$j_2:L\to M,$$ $$j_3:L\to V,$$ $$j_4:M\to L,$$ $$j_5:M\to M,$$ $$j_6:M\to V,$$ $$j_7:V\to L,$$ $$j_8:V\to M,$$ $$j_9:V\to V.$$ What I know: $j_9$ is impossible by Kunen's barrier.$j_1$ implies $M\neq L$,$j_5$ implies $M\neq V$. $j_6$ and $j_8$ are a puzzle for me.

Answer 1

For simplicity I work throughout in an appropriate class theory, say $\mathsf{NBG}$ or $\mathsf{MK}$. Also, I'll blackbox the fact that "There is a nonprincipal countably closed ultrafilter on some uncountable cardinal" is equivalent to "There is a measurable cardinal." This is a good exercise if you haven't seen it before! Also, Hamkins/Kirmayer/Perlmutter's Generalizations of the Kunen inconsistency is a good general source on this topic.

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First, note that since $\mathsf{V=L}$ is a first-order sentence, if either $A$ or $B$ is $L$ and $A$ is elementarily embeddable into $B$ then $A=B=L$. So $(1)$, $(2)$, and $(4)$ are equivalent to each other while $(3)$ and $(7)$ are outright inconsistent. Also, $(1)$ is equivalent to $(5)$ since if $j:M\rightarrow M$ is nontrivial elementary then so is $\hat{j}: L^M=L\rightarrow L^M=L$.

Next up we have $(6)$. When I originally wrote this answer I didn't know anything about this possibility, but today I ran into a very valuable source (see below). As earlier, an embedding $M\rightarrow V$ restricts to an embedding $L\rightarrow L$, so $(6)$ is at least as strong as $0^\sharp$. The obvious question (especially in light of the next section of this answer!) is the relationship between $(6)$ and a measurable cardinal. As it turns out, $(6)$ is rather weak, at least in consistency strength, unless one adds additional demands on the embedding or the source model $M$. See the beginning of Section $2$ of Vickers/Welch, On elementary embeddings from an inner model to the universe (or the rest of the paper more generally).

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Now we get to the fun stuff. The existence of an embedding of type $j_8$ is exactly equivalent to a measurable cardinal - and this is indeed the motivating observation which begins the whole study of elementary embeddings of inner models, so it is worth understanding well! Here's a sketch of the argument (due to Scott):

• One direction is easy: if $\mathcal{U}$ is a countably closed nonprincipal ultrafilter on an uncountable cardinal $\kappa$, then (utilizing Scott's trick to keep everything well-defined) we get an elementary embedding $\mathfrak{j}_\mathcal{U}$ from $V$ to the ultrapower $\prod V/\mathcal{U}$; by countable closure the latter is well-founded (and is easily seen to be set-like), and so it is definably-in-$V$ isomorphic to a unique inner model $M$. We usually conflate $\mathfrak{j}_\mathcal{U}$ with the induced elementary embedding of $V$ into this $M$.

• The other direction is a bit trickier: an arbitrary nontrivial elementary $j:V\rightarrow M$ might not "come from" an ultrafilter! However, it still implies the existence of an ultrafilter of the above type. Specifically, let $\kappa=crit(j)$ and consider $$\mathcal{U}_j:=\{A\subseteq\kappa: \kappa\in j(A)\}.$$ Then $\mathcal{U}_j$ is nonprincipal and countably closed.

One consequence of the above argument is that the a-priori-class-referring principle "There is a nontrivial elementary embedding of $V$ into some inner model $M$" is actually expressible in set theory alone. At the same time, it's important to note that we generally lose information in the $j\mapsto\mathcal{U}_j$ construction, in the sense that $\mathfrak{j}_{\mathcal{U}_j}\not=j$ in general. Figuring out how to "approximate" elementary embeddings by set-sized blobs of data is a general theme in inner model theory; for example, look at the notion of an extender, which is a generalization of an ultrafilter which lets us faithfully capture more embeddings.