## Luminy – Hugh Woodin: Ultimate L (II)

For the first lecture, see here.

II

In this lecture, we prove:

Universality Theorem. If $N$ is a weak extender model for $\mbox{}\delta$ is supercompact’, and $\pi:N\cap V_{\gamma+1}\to N\cap V_{\pi(\gamma)+1}$ is elementary with ${\rm cp}(\pi)\ge\delta$, then $\pi\in N$.

As mentioned before, this gives us that $N$ absorbs a significant amount of strength from $V$. For example:

Lemma. Suppose that $\kappa$ is 2-huge. Then, for each $A\subseteq V_\kappa,$ $(V_\kappa,\in,A)\models$There is a proper class of huge cardinals witnessed by embeddings that cohere $A$. $\Box$

Hence, if $A=N\cap V_\kappa$ and $\kappa>\delta$, then

$N\cap V_\kappa\models$There is a proper class of huge cardinals.

Here, coherence means the following: $j:V\to M$ coheres a set $A$ iff, letting $\gamma={\rm cp}(j)$, we have $V_{j(j(\gamma))+\omega}\subseteq M$ and $j(A)\cap V_{j(j(\gamma))+\omega}=A\cap V_{j(j(\gamma))+\omega}$. Actually, we need much less. We need something like $j''j(\gamma)\in M$ and for hugeness, $V_{j(j(\gamma))+1}\subseteq M$ already suffices.

This methodology breaks down past $\omega$-hugeness. Then we need to change the notion of coherence, since (for example, beginning with $j:L(V_\lambda)\to L(V_\lambda)$) to have $\pi:N\cap V_{\lambda+1}\to N\cap V_{\pi(\lambda)+1}$ is no longer a reasonable condition. But suitable modifications still work at this very high level.

The proof of the universality theorem builds on a reformulation of supercompactness in terms of extenders, due to Magidor:

Theorem (Magidor). The following are equivalent:

1. $\delta$ is supercompact.
2. For all $\gamma>\delta$ and all $a\in V_\gamma$, there are $\bar\delta<\bar\gamma<\delta$ and $\bar a\in V_{\bar\gamma}$, and an elementary $\pi:V_{\bar\gamma+1}\to V_{\gamma+1}$ such that:
1. $\bar\delta={\rm cp}(\pi)$ and $\pi(\bar\delta)=\delta$.
2. $\pi(\bar\gamma)=\gamma$ and $\pi(\bar a)=a$.

The proof is actually a straightforward reflection argument.

Proof. $(1.\Rightarrow 2.)$ Suppose that item 2. fails, as witnessed by $\gamma,a$. Pick a normal fine ${\mathcal U}$ on ${\mathcal P}_\delta(\lambda)$ where $\lambda=|V_{\gamma+1}|$, and consider

$j:V\to M\cong V^{{\mathcal P}_\delta(\lambda)}/{\mathcal U}$.

Then ${\rm cp}(j)=\delta$, $j(\delta)>\lambda$, and ${}^\lambda M\subseteq M$. But then ${}^{V_{\gamma+1}}M\subseteq M$, and, by elementarity, $j(\gamma),j(a)$ are counterexamples to item 2. in $M$ with respect to $j(\delta)$. However, $j\upharpoonright V_{\gamma+1}\in M$, and it witnesses item 2. in $M$ for $j(\gamma),j(a)$ with respect to $j(\delta)$. Contradiction.

$(2.\Rightarrow 1.)$ Assume item 2. For any $\lambda>\delta$ we need to find a normal fine ${\mathcal U}$ on ${\mathcal P}_\delta(\lambda)$. Fix $\lambda$, and let $\gamma=\lambda+\omega$ and $a=\lambda$. Let $\pi:V_{\bar\gamma+1}\to V_{\gamma+1}$ be an embedding as in item 2. for $\gamma,a$. Use $\pi$ to define a normal fine $\bar{\mathcal U}$ on ${\mathcal P}_{\bar\delta}(\bar\lambda)$ by

$A\in\bar{\mathcal U}$ iff $\pi''\bar\lambda\in\pi(A)$.

Note that $\pi''\bar\lambda\in V_{\lambda+1}\in V_{\gamma+1}$, so this definition makes sense. Further, ${\mathcal P}({\mathcal P}_{\bar\delta}(\bar\lambda))\in V_{\bar\lambda+2}\in V_{\bar\delta+1}$, so $\bar{\mathcal U}\in V_{\bar\gamma+1}$. Hence, $\bar{\mathcal U}$ is in the domain of $\pi$, and $\pi(\bar{\mathcal U})={\mathcal U}$ is as wanted. $\Box$

As mentioned in the previous lecture, it was expected for a while that Magidor’s reformulation would be the key to the construction of inner models for supercompactness, since it suggests which extenders need to be put in their sequence. Recent results indicate now that the construction should instead proceed directly with extenders derived from the normal fine measures. However, Magidor’s reformulation is very useful for the theory of weak extender models, thanks to the following fact, that can be seen as a strengthening of this reformulation:

Lemma. Suppose $N$ is a weak extender model for $\delta$ is supercompact’. Suppose $\gamma>\delta$ and $a\in V_\gamma$. Then there are $\bar \delta,\bar a,\bar\gamma$ in $V_{\delta}$ and an elementary $\pi:V_{\bar\gamma+1}\to V_{\gamma+1}$ such that:

1. ${\rm cp}(\pi)=\bar\delta$, $\pi(\bar\delta)=\delta$, $\pi(\bar\gamma)=\gamma$, and $\pi(\bar a)=a$.
2. $\pi(N\cap V_{\bar\gamma})=N\cap V_\gamma$.
3. $\pi\upharpoonright(N\cap V_{\bar\gamma+1})\in N$.

Again, the proof is a reflection argument as in Magidor’s theorem, but we need to work harder to ensure  items 2. and 3. The key is:

Claim. Suppose $\lambda>\delta$. Then there is a normal fine ${\mathcal U}$ on ${\mathcal P}_\delta(V_\lambda)$ such that

$\{x\in{\mathcal P}_\delta(V_\lambda)\mid$ The transitive collapse of $X\cap N$  is $N\cap V_{\bar\lambda}$, where $\bar\lambda$ is the transitive collapse of $X\cap\lambda\}\in{\mathcal U}$.

Proof. We may assume that ${}|V_\lambda|=\lambda$ and that this also holds in $N$. In $N$, pick a bijection $\rho$ between $\lambda$ and $N\cap V_\lambda$, and find ${\mathcal U}^*$ on ${\mathcal P}_\delta(\lambda)$ with ${\mathcal U}^*\cap N\in N$ and ${\mathcal P}_\delta(\lambda)\cap N\in{\mathcal U}^*$.

It is enough to check

$(*)$ $\{X\subseteq\lambda\mid$ The transitive collapse of $\rho[X]$ is a rank initial segment of $N\}\in{\mathcal U}^*$.

Once we have $(*)$, it is easy to use the bijection between $\lambda$ and $V_\lambda$ to obtain the desired measure ${\mathcal U}$.

To prove $(*)$, work in $N$, and note that the result is now trivial since, letting $j$ be the ultrapower embedding induced by the restriction of ${\mathcal U}^*$ to $N$, we have that $j''V_\lambda$ collapses to $V_\lambda$, which is an initial segment of $V$. $\Box$

Proof of the lemma. The argument is now a straightforward elaboration of the proof of Magidor’s theorem, using the claim just established. Namely, in the proof of $(1.\Rightarrow2.)$ of the Theorem, use an ultrafilter ${\mathcal U}$ as in the claim. We need to see that (the restriction to $V_{\gamma+1}$ of) the ultrapower embedding $j$ satisfies $j(N\cap V_{\bar\gamma})=N\cap V_\gamma$. We begin with $\lambda$ much larger than $\gamma$ such that $\lambda=|V_\lambda|$, and fix sets $A,B\in N\cap{\mathcal P}(\lambda)$ such that $|A|=|B|=\lambda$, and a bijection $\rho:\lambda\to V_\lambda$ such that $\rho\upharpoonright A$ is a bijection between $A$ and $N\cap V_\lambda$ and $\rho\upharpoonright A\in N$.

We use $\rho$ to transfer ${\mathcal U}$ to a measure ${\mathcal U}^*$ on ${\mathcal P}_\delta(A)$ concentrating on $N$. Now let $j:V\to M\cong V^{{\mathcal P}_\delta(A)}/{\mathcal U}^*$ be the ultrapower embedding. We need to check that $j(N)\cap V_\lambda=N\cap V_\lambda$. The issue is that, in principle, $j(N)\cap V_\lambda$ could overspill and be larger. However, since ${\mathcal U}^*$ concentrates on $N$, this is not possible, because transitive collapses are computed the same way in $M$, $N$, and $V$, even though $(N^{{\mathcal P}_\delta(A)}/{\mathcal U}^*)^N$ may differ from $(N^{{\mathcal P}_\delta(A)}/{\mathcal U}^*)^V$. $\Box$

We are ready for the main result of this lecture.

Proof of the Universality Theorem. We will actually prove that for all cardinals $\gamma>\delta$, if

$\pi:(H(\gamma^+))^N\to(H(\pi(\gamma)^+))^N$

is elementary, and ${\rm cp}(\pi)\ge\delta$, then $\pi\in N$.

This gives the result as stated, through some coding.

Choose $\lambda$ much larger than $\gamma$, and let $a=(\pi,\gamma)$. Apply the strengthened Magidor reformulation, to obtain $\bar a=(\bar\pi,\bar\gamma)$, $\bar\lambda$ and $\bar\delta$, and an embedding

$\sigma:V_{\bar\lambda+1}\to V_{\lambda+1}$

with ${\rm cp}(\sigma)=\bar\delta$, $\sigma(\bar\delta)=\delta$, $\sigma(\bar\pi)=\pi$, and $\sigma(\bar\gamma)=\gamma$.

Note that $\bar\pi: H(\bar\gamma^+)^N\to H(\bar \pi(\bar \gamma)^+)^N$.

It is enough to show that $\bar\pi\in N$, since $\sigma\upharpoonright (V_{\bar\lambda+1}\cap N)\in N$, and so $\pi=\sigma(\bar\pi)\in N$ as well.

For this, we actually only need to show that $\bar\pi\upharpoonright({\mathcal P}(\bar\gamma)\cap N)\in N$, since the fragment $\pi\upharpoonright({\mathcal P}(\gamma)\cap N)$ of $\pi$ determines $\pi$ completely. The advantage, of course, is that it is easier to analyze sets of ordinals.

Let $a\subseteq\bar\gamma$ with $a \in N$, and let $\alpha\in\pi(\bar\gamma)$. We need to compute in $N$ whether $\alpha\in\bar\pi(a)$. For this, note that

$\alpha\in\bar\pi(a)$ iff $\sigma(\alpha)\in\sigma(\bar\pi(a))$.

Now, $\sigma(\bar\pi)=\pi$, so this reduces to $\sigma(\alpha)\in\pi(\sigma(a))$, i.e., to compute $\pi$, it suffices to know $\pi\upharpoonright{\rm ran}(\sigma)$.

Recall that $\sigma\upharpoonright N\cap V_{\bar\lambda+1}\in N$, and consider $\sigma_0=\sigma\upharpoonright H(\bar\gamma^+)^N$. Note that $\sigma_0\in H(\gamma^+)^N$, and $\sigma_0:H(\bar\gamma^+)^N\to H(\gamma^+)^N$. Applying $\pi$ to $\sigma_0$, and using elementarity, we have

$\pi(\sigma_0):\pi(H(\bar\gamma^+)^N)\to\pi(H(\gamma^+)^N)$.

But $\pi(H(\bar\gamma^+)^N)=H(\bar\gamma^+)^N,$ because ${\rm cp}(\pi)\ge\delta$, while $\pi(H(\gamma^+)^N)=H(\pi(\gamma)^+)^N$.

It follows that $\pi(\sigma(a))=\pi(\sigma_0)(\pi(a))=\pi(\sigma_0)(a)$. Since $\sigma_0\in {\rm dom}(\pi)$, we have $\pi(\sigma_0)\in N$ (simply note the range of $\pi$), and we are done, because we have reduced the question of whether $\alpha\in\bar \pi(a)$ to the question of whether $\sigma(\alpha)\in\pi(\sigma_0)(a)$, which $N$ can determine. $\Box$

Note how the Universality Theorem suggests that the construction of $L[\vec E]$ models for supercompactness using Magidor’s reformulation runs into difficulties; namely, if $\delta$ is supercompact, we have many extenders $F$ with critical point $\kappa_F<\delta$ and $\pi_F(\kappa_F)=\delta$, and we are now producing new extenders above $\delta$, that should somehow also be accounted for in $\vec E$.

A nice application of universality is the dichotomy theorem for ${\sf HOD}$ mentioned at the end of last lecture. If ${\sf HOD}$ is a weak extender model for supercompactness, we obtain the following:

Corollary. There is no sequence of (non-trivial) elementary embeddings ${\sf HOD}\to_{j_0}{\sf HOD}\to_{j_1}{\sf HOD}\to_{j_2}\dots$ with well-founded limit. $\Box$

It follows that there is a $\Sigma_2$-definable ordinal such that any embedding fixing this ordinal is the identity! This is because ${\sf HOD}=L[T]$ where $T$ is the $\Sigma_2$-theory in $V$ of the ordinals.

In particular, there is no $j:({\sf HOD},T)\to({\sf HOD},T)$. Note that the corollary and this fact fail if ${\sf HOD}$ is replaced by an arbitrary weak extender model.

The question of whether there can actually be embeddings $j:{\sf HOD}\to{\sf HOD}$ in a sense is still open, i.e., its consistency has currently only been established from the assumption in ${\sf ZF}$ that there are very strong versions of Reinhardt cardinals, i.e., strong versions of embeddings $j:V\to V$, the consistency of which is in itself problematic.

(On the other hand, Hugh has shown that there are no embeddings $j:V\to{\sf HOD}$, and this can be established by an easy variant of Hugh’s proof of Kunen’s theorem as presented, for example, in Kanamori’s book (Second proof of Theorem 23.12).)

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