Set forcing and ultrapowers

Question

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?)

Answer 1

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.

Answer 2

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."