From Determinacy to a Woodin II

In the prequel I sketched a proof of how determinacy hypotheses could imply the measurability of both $(\bf\delta^1_1)^{L(\mathbb R)}$ and $(\bf\delta^2_1)^{L(\mathbb R)}$ inside $L(\mathbb R)$. The latter is really the first step in showing the much stronger assertion that $\Theta^{L(\mathbb R)}$ is Woodin. I’ll here sketch what main ideas are involved in the proof of this fact.

This post is part of a series on determinacy:

The main theorem in question is thus the following, due to Woodin.

Theorem (Woodin). Assume ZF+DC+AD. Then
$$ \text{HOD}^{L(\mathbb R)}\models\Theta^{L(\mathbb R)}\text{ is a Woodin cardinal}. $$

I should note that the theorem still holds without the use of DC, by using an alternative characterisation of Woodin cardinals - but I’ll stick the following one.

Definition. A cardinal $\delta$ is Woodin if for every $A\subseteq\delta$ there exists a cardinal $\kappa<\delta$ which is A-reflecting, which is to say that given any $\lambda\in(\kappa,\delta)$ we can find an elementary embedding $j:V\to M$ with critical point $\kappa$ and satisfying $j(\kappa)>\lambda$, $V_\lambda\subseteq M$ and $A\cap V_\lambda=j(A)\cap V_\lambda$.

To show that $\Theta:=\Theta^{L(\mathbb R)}$ is Woodin in HOD we can first of all focus on the case where $A=\emptyset$, meaning that we need to find an $\emptyset$-reflecting $\kappa<\Theta$ – this will be the case we focus on here in this post, as the case for arbitrary $A$ turns out to be a relativisation of this case. This $\kappa$ will turn out to be precisely $\bf\delta^2_1:=(\bf\delta^2_1)^{L(\mathbb R)}$, so to every $\lambda<\Theta$ we need to find an elementary embedding $j:V\to M$ with critical point $\bf\delta^2_1$, $j(\bf\delta^2_1)>\lambda$ and $V_\lambda\subseteq M$.

The main new gadget that we’re going to use is a reflection phenomenon at $\bf\delta^2_1$: there exists a function $F:{\bf\delta^2_1}\to L_{\bf\delta^2_1}(\mathbb R)$ such that

Given any $X\in L(\mathbb R)\cap\text{OD}^{L(\mathbb R)}$, $z\in{^\omega\omega}$ and $\Sigma_1$ formula $\varphi$, if

$$ L(\mathbb R)\models\varphi[z,X,{\bf\delta^2_1},\mathbb R] $$

then there’s a $\delta<\bf\delta^2_1$ such that

$$ L(\mathbb R)\models\varphi[z,F(\delta),\delta,\mathbb R]. $$

This $F$ is constructed analogously to $\diamondsuit$-sequences in $L$, i.e. defining it by least counterexample. To any $X$ as above we pick a universal $\Sigma_1^{L(\mathbb R)}(\{X,{\bf\delta^2_1},\mathbb R\})$ set $U_X$ and for each $\delta<\bf\delta^2_1$ let $U_\delta$ be a universal $\Sigma_1^{L(\mathbb R)}(\{F(\delta),\delta,\mathbb R\})$ set, obtained by using the same definition as $U_X$. We now claim that to each $\Sigma_1$-formula $\varphi$ and real $y\in{^\omega\omega}$ we can construct a real $z_{\varphi,y}$ such that

$$ z_{\varphi,y}\in U_X\text{ iff }L(\mathbb R)\models\varphi[y,X,{\bf\delta^2_1},\mathbb R]. $$

To define the $z_{\varphi,y}$ note that the right-hand side above is a $\Sigma_1^{L(\mathbb R)}({X,{\bf\delta^2_1},\mathbb R})$ formula, meaning that the set of $y$'s satisfying it is in $(U_X)_x$ for some $x\in{^\omega\omega}$, so that we can define $z_{\varphi,y}:=\langle x,y\rangle$. We can then reformulate the above reflection phenomenon as $U_X\subseteq\bigcup_{\delta<{\bf\delta^2_1}}U_\delta$.

Now to actually define our measure on $\bf\delta^2_1$. Let $\lambda<\Theta$ be arbitrary and fix an OD pre-wellordering $\leq_\lambda$ of the reals of order-type $\lambda$. Then our desired $X$ is going to be $X:=(\leq_\lambda,\lambda)$. To each $S\subseteq\bf\delta^2_1$ we can now define the game $G^X(S)$ as

Here the only rule is that $(x)_i,(y)_i\in U_X$ for every $i<\omega$. In this case that very rule can be seen as a $\Sigma_1^{L(\mathbb R)}(\{X,{\bf\delta^2_1},\mathbb R\})$ statement, so the reflection phenomenon applies and there is some $\delta<\bf\delta^2_1$ such that $(x)_i,(y)_i\in U_\delta$ for every $i<\omega$. Then player I wins iff $\delta\in S$. Analogously to the previous measures we then set

$$ \mu_X:=\{S\subseteq{\bf\delta^2_1}\mid\text{Player I wins }G^X(S)\}. $$

Now, what’s special about this measure as opposed to the measure we found in my previous post? The key set is $S_0$, consisting of all $\delta<\bf\delta^2_1$ such that $F(\delta)=(\leq_\delta,\lambda_\delta)$, where $\leq_\delta$ is a pre-wellordering of the reals of order-type $\lambda_\delta$ and that $L_{\lambda_\delta}(\mathbb R)$ satisfies a sufficient chunk of ZFC.

Let $Q_\alpha^\delta$ ($Q_\alpha$) be the $\alpha$'th component of $\leq_\delta$ ($\leq_\lambda$). Furthermore, for every $\delta\in S_0$ and $t\in{^\omega\omega}$ let $\alpha_t^\delta$ be the unique $\alpha$ such that $t\in Q_\alpha^\delta$ and define the functions $f_t:S_0\to\bf\delta^2_1$ as $f_t(\delta):=\alpha_t^\delta$. A major theorem is that $[f_t]_{\mu_X}$ collapses to the $\leq_\lambda$-rank of $t$ in the ultrapower. This means straight away that $\lambda < j_X({\bf\delta^2_1})$ as $f_t(\delta)<\bf\delta^2_1$ for every $t$ and $\delta$.

The $\lambda$-strongness of $j$ is shown by describing any subset $A\subseteq\lambda$ with $A\in\text{HOD}^{L(\mathbb R)}$ in terms of the $Q_\alpha$'s using a new coding lemma, so that we get a “reflected version” $A^\delta$ of $A$, which we can use to describe $A$ in the ultrapower. See (some) details in my note.