# The Stationary Tower II – the construction

In the last post we developed the machinery of generic ultrapowers, which enabled us to go from a uniform normal ideal in $V$ to getting an elementary embedding $j:V\to\mathcal M$ with $\text{crit }j=\omega_1^V$ lying in a generic extension of $V$if we assume that we have a Woodin cardinal. The model $\mathcal M$ furthermore enjoyed the property of being closed under countable sequences in the forcing extension. We now introduce a generalisation of such an embedding, which grants more closure — this is precisely given by the stationary tower.

So, say we want to construct a forcing notion $\mathbb P$ such that a generic $G$ for $\mathbb P$ induces an elementary embedding $j:V\to M$ satisfying that ${^{<\delta}}M\subseteq M$. If we focus on a particular $\lambda<\delta$, we essentially want to show that $\kappa:=\text{crit} j$ is “$V$-supercompact” in $V[G]$.

To witness this we need a normal fine ultrafilter over $\mathcal P_\kappa(\lambda)$, so our forcing notion should approximate such measures for every $\lambda<\delta$ simultaneously. Since we want our resulting measures to be normal, we’re not interested in considering non-stationary sets. A naive attempt would then be to define

$\mathbb P:=\bigcup_{\lambda<\delta}\{s\in\mathcal P_\kappa(\lambda)\mid s\text{ is stationary}\}$.

What should the ordering be? We at least want to connect stationary subsets of $\mathcal P_\kappa(\lambda)$ with stationary subsets of $\mathcal P_\kappa(\theta)$ for every $\lambda,\theta<\delta$. If $s\in\mathcal P_\kappa(\lambda)$ is stationary then $\bigcup s=\lambda$, so we don’t really need to keep track of the $\lambda$‘s as well. The first condition is then that for $s,p\in\mathbb P$,

$s\leq p\Rightarrow \bigcup p\subseteq\bigcup s$.

We want something more than this though, as otherwise everything would be comparable. For, say, $p,s\in\mathbb P$ such that $\bigcup p=\bigcup s$, we would want that $p\leq s$ iff $s\subseteq p$. This leads us to the following definition.

Definition. The $\kappa$-stationary tower for some uncountable $\kappa$, associated to a Woodin cardinal $\delta>\kappa$, is the poset

$\mathbb P:=\{s\in V_\delta\mid s\text{ is stationary over } \mathcal P_\kappa(\bigcup s)\}$,

where $s\leq p$ iff $\bigcup p\subseteq\bigcup s$ and $\{A\cap\bigcup p\mid A\in s\}\subseteq p$.

The $\delta$-stationary tower is denoted by $\mathbb P_{<\delta}$ and called the full stationary tower, and the $\omega_1$-stationary tower is denoted by $\mathbb Q_{<\delta}$ and called the countable stationary tower.

So, let’s see that this actually works. I’ll focus on the full stationary tower, but the same proof works for any other variant as well. We need to find a $V$-extender in $V[G]$ witnessing the existence of a $j:V\to M$ such that $M$ is closed under sequences in $V[G]$ of length $<\delta$. To each $X\in V_\delta-\{\emptyset\}$ we define a measure $\mu_X:\mathcal P(\mathcal P(X))\to 2$ given by

$\mu_X(s)=1$ iff $s=\{X\cap A\mid A\in p\}$ for some $p\in G$ such that $X\subseteq\bigcup p$.

Proposition. $\mu_X$ is a $V$-measure over $\mathcal P(X)$, for every non-empty $X\in V_\delta$.

Proof. That $\mu_X$ is a filter over $\mathcal P(X)$ follows from $G$ being a filter over $\mathbb P_{<\delta}$. To show that $\mu_X$ is an ultrafilter, let $A\subseteq\mathcal P(X)$ be any set. We then want to show that

$\{p\in\mathbb P_{<\delta}\mid p_X\subseteq A\lor p_X\cap A=\emptyset\}$ is dense.

Let therefore $p\in\mathbb P_{<\delta}$ be arbitrary and define the two sets

$p^0:=\{x\in p\mid x\cap X\in A\}$ and $p^1:=\{x\in p\mid x\cap X\notin A\}$.

Then as $p^0\cup p^1=p$ is stationary, one of the two is stationary, say $p^0$. Then we’re done, since $p^0\leq p$ and $p^0_X\subseteq A$. QED

That $\mu_X$ is actually $V$-normal and $V$-fine, i.e. that the corresponding embedding $j:V\to\mathcal M$ satisfies that $\mathcal M$ is closed under $|X|$-sequences in $V[G]$, is quite a long proof and I won’t give it here. The proof goes indirectly by showing that if $\delta$ satisfies that given any $|X|$-sequence of predense sets of $\mathbb P_{<\delta}$ we can find an inaccessible $\gamma<\delta$ such that every predense set restricted to $\mathbb P_{<\gamma}$ is so-called semi-proper, then $\mathcal M$ is closed under $|X|$-sequences. It turns out that when $\delta$ is Woodin, this condition is satisfied.

We still need to show that all the resulting ultrapowers give rise to a single ultrapower, giving us closure under all sequences of length $<\delta$ simultaneously. In other words, we need a directed system, just like we have with extenders. But this turns out to be straightforward, using the following lemma, which follows just by going through the definitions.

Lemma. Let $X\subseteq Y$ and $X\neq\emptyset$.

1. If $p\subseteq\mathcal P(X)$ is stationary then $\{A\subseteq Y\mid A\cap X\in p\}$ is stationary.
2. $\mu_X(p)=1$ iff $\mu_Y(\{A\subseteq Y\mid A\cap X\in p\})=1$.

We can then for $X\subseteq Y$ define maps $i_{XY}:\text{Ult}(V,\mu_X)\to\text{Ult}(V,\mu_Y)$ as $i_{XY}([f]_{\mu_X}):=[f^Y]_{\mu_Y}$, where $f^Y(A):=f(A\cap X)$. This grants us with a directed system, so taking the direct limit of the ultrapower we wind up with our elementary $j:V\to\mathcal M$ in the generic extension such that $\mathcal M$ is closed under $<\delta$-sequences in $V[G]$.

One neat feature is that we can more or less choose the critical point of $j:V\to\mathcal M$ to be anything we want. More specifically, we can set it to be any uncountable regular cardinal $\kappa<\delta$. To see this, note first that $\kappa$ is stationary in $\mathcal P(\kappa)$, so that $\kappa\in\mathbb P_{<\delta}$. Now let $g\subseteq\mathbb P_{<\delta}$ be $V$-generic such that $\kappa\in G$. Analogously to measures $\mu$ where $x\in\mu$ iff $\text{crit }\mu\in j_\mu(x)$, we get that $j"\kappa\in j(\kappa)$. This means that $j"\kappa$ is an ordinal, so that $j"\kappa=\kappa$, and furthermore that $\kappa, making it the critical point of $j$.

Summing up, we more or less did the same thing as with the generic ultrapowers, but we needed a way to go from only a single generic to get ultrapowers at every “level”, i.e. on every $\mathcal P(X)$, which were coherent. Now, given this new tool at hand, what can we use it for? It turns out that the stationary tower forcing and the genericity iterations seem to give “dual” proofs to various applications. I’ll get back to that later.

For now, all there is left to say is: merry Christmas everyone!