From mice to determinacy

Last time we proved that mice M with n<\omega Woodins knows about \bf\Sigma^1_{n+1} sets A, meaning A\cap M\in M, using Woodin’s genericity iterations and the notion of mice understanding sets of reals. But what good is a projectively aware mouse? To give an example of the usefulness of this property, we show that the existence of these projectively aware mice yields determinacy of sets of reals, shown by Neeman (’02).

Before we do that, however, let’s abstract away from the Woodins. We want to isolate the key features of the genericity iterations, and the following notion does exactly that.

Definition. Let M be a mouse and \eta\in o(M). Then we say that M absorbs reals at \eta if whenever \xi<\eta, x\in\mathbb R and i:M\to N is an iteration below \xi then there exists an iteration j:N\to P below i(\eta) with \text{crit} j>i(\xi) and x\in P[g] for some P-generic g\subseteq\text{Col}(\omega,j(i(\eta))).

By using genericity iterations we then get that M will always absorb reals at its Woodin cardinals. This is even close to being an equivalence: if M\models\textsf{ZFC}^-+\delta^+\text{exists} and if M absorbs reals at \delta then either \delta is Woodin or a limit of Woodins in M.

Now, recall from our last post that a mouse M understands a set of reals at \eta<o(M) if there’s a term \tau such that whenever P is an iterate of M and g\subseteq\text{Col}(\omega,i(\eta)) is P-generic then A\cap P[g]=i(\tau)^g. We now also say that M captures A at \eta if M absorbs reals at \eta and understands A at \eta, and say that M Suslin-understands (Suslin-captures) A at \eta if the corresponding forcing term witnessing it is of the form p[\check T] for a tree T\in M on some \omega\times\kappa.

Our theorem from last time then says that whenever M captures B\subseteq\mathbb R^2 at some \eta then M understands \exists^{\mathbb R}B at every \xi<\eta. To yield determinacy from this we have to require M to have some more specialised knowledge of the set of reals in question.

Theorem. Let A be a set of reals and assume that there’s a countable mouse M Suslin-capturing both A and \lnot A, where the trees witnessing this are homogeneous in M. Then A is determined.

Proof. Let \tau=p[\check T] and \tau'=p[\check S] witness that M Suslin-captures A and \lnot A at \eta, respectively. Since T is homogenous in M it thinks that p[T] is determined, using the Martin-Steel theorem that we’ve covered in a previous post. Let \sigma\in M be a winning strategy, say for player I without loss of generality.

Assume A is not determined, so that there’s a play y\in\mathbb R following \sigma, but where player I loses; i.e. that x\notin A. Use that M absorbs reals at \eta to yield an iteration i:M\to N with y\in N[g] for some N-generic g\subseteq\text{Col}(\omega,\eta)^N. But since M understands A and \lnot A at \eta we get that i(\tau)^g=A\cap N[g] and i(\tau')^g=\lnot A\cap N[g]. This means that that y\in i(\tau')=p[i(S)] by definition of y, so that

N[g]\models\exists x\in p[i(S)]: x\text{ follows }i(\sigma),

which makes sense as i(S) and i(\sigma) are elements of N[g]. But then absoluteness of wellfoundedness yields that this is true in N as well, so elementarity of i then contradicts that \sigma is winning in M. QED

The clause concerning the homogeneity of the trees may seem a bit forced, but recall the Martin-Steel result from our previous post that if \delta_0<\delta_1 are Woodin cardinals of M then whenever T,S\in M project to complements after forcing with \text{Col}(\omega,\delta_1), they’re homogeneous in M. This means that we get the following corollary.

Corollary. Let A be a set of reals and let M be a countable mouse with Woodins \delta_0<\delta_1. Assume that M Suslin-captures both A and \lnot A at \delta_1. Then A is determined.

The Woodins here are really used in their full force, and not simply a way to get genericity iterations, so it seems that we have to move to the more ‘concrete’ Woodin cardinals rather than only assuming that M absorbs reals. This corollary might still seem a bit niche, but this is precisely the result which is used again and again in the core model induction to yield determinacy results. More on that some other time.