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.

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  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

Capture

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.