Characterisation of Woodins in mice

I’ve previously mentioned an interesting characterisation of Woodin cardinals, that they are more or less characterised by the fact that we can do genericity iterations with them, i.e., that we can “capture reals” when Woodin cardinals are present. This is an exercise in the core model induction book, and Stefan Mesken recently found a way to solve this (see his solution here), which I’ll be presenting here.

I’ve previously introduced the notion of absorbing reals, but let’s recall it here for convenience:

Definition. Let M be a premouse, \Sigma an iteration strategy for M and \delta\in o(M). Then we say that (M,\Sigma) absorbs reals at (\eta, \delta) if there exists an iteration i:N\to P by \Sigma living on (\eta, \delta) such that x\in P[g] for some P-generic g\subseteq\text{Col}(\omega,i(\delta)). We also say that (M,\Sigma) absorbs reals at \delta if it absorbs reals at (\eta,\delta) for all \eta<\delta.

The characterisation is then this, which is Exercise 1.4.5 in the core model induction book:

Proposition. Let M be a premouse, \delta\in o(M) such that M\models\textsf{ZFC}^-+\delta^+ exists. Then (M,\Sigma) absorbs reals at \delta iff \delta is either a Woodin cardinal in M or a limit of Woodin cardinals in M.

Proof. The backwards direction is by using genericity iterations, which I’ve previously covered, so we show the forwards direction. Firstly we may assume that M is countable, as otherwise we could take a countable hull N:=\text{cHull}^M(\{\delta\}) and work with N and its pulled back version of \Sigma.

Assume that \delta is not Woodin in M; we’ll show that it’s a limit of Woodins in M. Let \eta<\delta be an ordinal and define \delta^*\in(\eta,\delta] to be the least ordinal which absorbs reals at (\eta,\delta). We will show that \delta^* is in fact Woodin in M, so assume it’s not. Since Woodins always absorb reals we get that there are no Woodins in the interval (\eta,\delta^*] in M.

As M doesn’t have any Woodins in the interval (\eta,\delta^*] we get that any \Sigma-iteration living on (\eta,\delta^*) is guided by Q-structures, so that M can define \Sigma as

\Sigma(\mathcal T)=b iff \text{Col}(\omega,\mathcal T)^M\Vdash\mathcal Q(\check b,\check{\mathcal T})\text{ exists and }\mathcal M(\check{\mathcal T})\trianglelefteq\mathcal Q(\check b,\check{\mathcal T}).

For more information about Q-structures, see e.g. John Steel’s handbook chapter or my MSc thesis. The reason why the Q-structures exist in the generic extension is by using absoluteness of wellfoundedness.

Screenshot 2019-02-27 at 19.40.42

Now let \mathcal T be an iteration tree on M living on (\eta,\delta^*) with last model P such that M\in P[g] with g\subseteq\text{Col}(\omega, i^{\mathcal T}(\delta^*)) P-generic. Then i\upharpoonright (\delta^*)^{+M}\in M\subseteq P[g] by the above. But i\upharpoonright(\delta^*)^{+M} is cofinal in i(\delta^*)^{+P}, making it singular in P[g]. But it’s regular in P and \text{Col}(\omega,i(\delta^*))^P has the i(\delta^*)^{+P}-cc, so it’s also regular in P[g], a contradiction! So \delta^* is Woodin in M, and \delta is thus a limit of Woodins in M. QED