# Set forcing and ultrapowers

The following is a result of Woodin (the proof is found after Theorem 5 of "Generalizations of the Kunen Inconsistency" by J.D.Hamkins, G.Kirmayer and N.L.Perlmutter):

(Woodin) Let $V[G]$ be a set-forcing extension of $V$. Then there is no non-trivial elementary embedding $$j:V[G]\prec V.$$ Corollary 6 gives the theorem from the perspective of the extension as: If $$j:V\prec M$$ is a non-trivial elementary embedding in $V$, then $M$ is not a set-forcing extension of $V$.

From the point-of-view of the generic extension, the corollary can be read as something like "I am not an ultrapower of the ground model by $U$." This must be true for every ultrafilter $U$ in the generic extension. Since the generic extension was an arbitrary set-forcing extension, it seems to me that the corollary implies that ultrapowers of $V$ (or maybe I should say "transitive collapses of ultrapowers of $V$"?) are not obtainable from set forcing (over $V$).

If this is true, I wonder if this was known before the proof presented in the "Generalizations..." paper and even if there is another, substantially different proof (whatever this could mean).

Further (and perhaps this is a silly question with an obvious answer) can ultrapowers of $V$ be obtained by class forcing? Given a transitive set/class $M$, could forcing over $M$ (where the p.o. is considered a class from the point-of-view of $M$) yield a set/class which is isomorphic to some (all?) ultrapower of $M$?

(Added later: I take Woodin's original result to say something like "the ground model is not the transitive collapse of any ultrapower by $U$" is true from the point-of-view of a set-forcing extension's point-of-view. Is this understanding correct?)

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Do you mean $j: V \to V[G]$? –  Monroe Eskew Feb 20 '13 at 22:18
–  Asaf Karagila Feb 20 '13 at 22:29
Everett, could you clarify whether in your questions about adding ultrapowers by class forcing, do you really mean ultrapowers, or are you referring to any target model of an elementary embedding of $V$? –  Joel David Hamkins Feb 21 '13 at 1:30
It seems clear that no forcing extension of $V$ can yield an ultrapower of $V$ by an ultrafilter in $V$: we can assume WLOG the ultrafilter in question is countably closed, since forcing extensions preserve well-foundedness, and any ultraproduct of $V$ is isomorphic to a transitive subclass of $V$, which is not true of any nontrivial forcing extension of $V$. I think with some more work, we can extend this to ultrafilters not in $V$, but I'm not sure. Or maybe I'm missing something? –  Noah Schweber Feb 21 '13 at 1:33
Everett, the second author's name is Greg Kirmayer, with only one K. –  Joel David Hamkins Feb 21 '13 at 3:03

The paper you refer to is available here: Generalizations of the Kunen inconsistency.

In your question, it seems that things got switched around a little when you went from the theorem to the corollary, since in the paper Corollary 6 asserts that if $j:V\to M$ is an elementary embedding, then $V$ is not a forcing extension of $M$. (But your question asserts that $M$ is not a forcing extension of $V$.) So this corollary is simply an equivalent formulation of Woodin's theorem that there is no $j:V[G] \to V$, expressed from the perspective of the forcing extension rather than from the ground model. Theorem 5 and corollary 6 are equivalent ways of expressing the same fact.

Nevertheless, your formulation of the corollary happens to be a true statement, because this is a similar re-casting of theorem 7 of the paper, which asserts that there is no nontrivial elementary embbedding $j:V\to V[G]$. That is, what we've got is no nontrivial $j:V\to V[G]$ and also no nontrivial $j:V[G]\to V$. From the perspective of the extension, what this says is that if $j:V\to M$, then $M$ is not a forcing extension of $V$, as you say.

Both of these results are unified by the claim (theorem 8) that there is no nontrivial elementary embedding $j:M\to N$ whenever $M$ and $N$ are grounds of $V$. And this fact is a consequence of the more general claim in the paper that there is no $j:M\to N$ whenever $M$ and $N$ are eventually stationary correct. This is theorem 10 of the paper, and it provides a partial answer to your question about class forcing, since many of the usual class forcings that one might undertake are eventually stationary correct.

You ask about the history, and this is a little less clear. This is what we say in the paper:

Attribution for this next theorem [theorem 7] is not clear to us. Woodin reportedly proved it along with theorem 5 while he was a graduate student in the early 1980s. But also, Matt Foreman mentioned to the ﬁrst author [Hamkins] that he discussed a version of the theorem with Mack Stanley and Sy Friedman around the same time, but their proof was evidently different than ours, and unfortunately the result was not published. Suzuki proved a theorem implying our theorem 7 in [Suz98, p. 344], using a technique essentially the same as ours. (Suzuki proved that if there is $j : V \to M$ in $V[G]$, then $V\not\subset M$. This result is stronger than our theorem 7, but weaker than our theorem 10. Although Suzuki states in his introduction that his proof only concerns definable $j$, in fact his proof never uses that fact and can be formalized in NGBC.)

I don't know the Foreman/Friedman proof, but this may be the alternative proof you seek. In any case, it seems that Suzuki may have the first proof of that fact to appear in print.

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On my latest reading I have become confused. I have a sense that there is/was no distinction between an elementary embedding $j:V\prec V[G]$ and an elementary embedding $j:V[G]\prec V$ where $G$ is a $V$-generic subset of a set $\mathbb{P}$. I guess I'm still trying to understand the nuance here. But thank you for pointing out the historical relevance of your remarks. –  Everett Piper Feb 21 '13 at 5:16
The case of $j:V[G]\to V$ is seemingly like the normal situation of a large cardinal embedding, but inside $V[G]$, for one has an elementary embedding from the entire universe, $V[G]$, into a transitive class $V$. The case of $j:V\to V[G]$, in contrast, is a bit stranger, even from the perspective of $V[G]$, since it maps a transitive class into the whole universe. Later on in the paper we rule out $j:V\to \text{HOD}$ and also $j:\text{HOD}\to V$, which have a similar appearance, but the arguments in these cases are totally different. –  Joel David Hamkins Feb 21 '13 at 11:02

In section 2.3 of Paul Larson's book, The Stationary Tower, he shows that if $V$ has a proper class of completely Jonsson cardinals, then forcing with the class-sized stationary tower $\mathbb{P}_\infty$ yields an elementary embedding $j : V \to V[G]$. Here, $V[G]$ is a direct limit of generic ultrapowers. He notes that $V$ and $G$ are not definable classes in $V[G]$, so this embedding only exists "from the outside."

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