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Shelah Product Tree Forcing

This forcing is used to prove the independence of the number of Ramsey ultrafilters from \mathrm{ZFC}. We will give a brief overview of the results discussed in Halbeisen’s book.  To be precise:

Theorem: Let 0 \le \kappa \le \omega_2 be any cardinal. Then there is a model of \mathrm{ZFC} with exactly \kappa Ramsey ultrafilters.

The idea behind the proof is to start with a lot of Ramsey ultrafilters, say \omega_2 many, for instance by starting in a model where \mathrm{CH} + 2^{\mathfrak{c}} = \omega_2 holds. Then we have that \mathrm{CH} implies \mathrm{MA}, which in turn can be  used to construct \omega_2 = 2^\mathfrak{c} RUF. Next, we choose \kappa of them to preserve, and destroy all of the others. This can be done with an iteration of some forcing notion whose job it is to destroy exactly one RUF. One then has to make sure of course that:

  • The  iterated forcing notions don’t interfere with each other.
  • No new RUF are added at limit stages.

This forcing notion which will be responsible for eliminating a RUF at a time is Shelah’s product tree forcing. First some preliminary definitions.

    \[ T_n^\otimes = \prod_{0\le l \le n} {}^l2 \quad T^\otimes = \bigcap_{n\in \omega} T_n^\otimes\]

We define for \eta, \zeta \in T^\otimes that \zeta extends \eta if: 

    \[ \eta \prec \zeta \Leftrightarrow \eta \in T_n^\otimes \wedge \zeta \in T_m^\otimes \wedge n \le m \wedge \zeta|_{n+1} = \eta \]

We say a subset T \subset T^\otimes is a (product) tree if for all \zeta\in T and \eta \in T^\otimes such that \eta \prec \zeta we have \eta \in T, i.e. T is closed under taking initial segments.  A tree is called perfect, if it splits indefinitely, i.e. for each element, we can find two different extensions.  Product trees are a little more complicated than the usual trees encountered in Sacks or Miller forcing, since at level n, they potentially split into 2^n branches. Let’s formalize that idea.

    \[ T_n = T \cap T_n^\otimes \]

is called the restriction of T up to level n.  If we have an element \eta \in T_n and some t\in {}^n2, then we can concatenate them to get an element \eta ^\frown t \in T_{n+1}. Let \eta \in T be given. Then:

    \[ \Theta_\eta = \{t \in {}^n2: \eta \frown t \in T_{n+1} \}\]

Now we are ready to define the n-th meta level. 

    \[ T \llbracket n\rrbracket = \{ \Theta_\eta : \eta \in T_n\} \]

Next, we say a binary tree t \subset {}^n2 is a k-tree, if it is a full binary tree when restricted to its first k levels. We can then look at how “branching” the levels of T are by examining: 

    \[ \text{fbt}_k(T) = \{ n \in \omega : \forall \Theta_\eta \in T \llbracket n\rrbracket \forall t \in\Theta_\eta (t \text{ is a k-tree})\}\]

So in other words: n \in\text{fbt}_k(T) means that every node on the n-th meta level branches into at least 2^k nodes on the next meta-level. We are now finally in good shape to define the Shelah tree forcing. Consider a fixed P-family \mathcal{V}. The conditions T_\mathcal{V}^\otimes of the Shelah tree forcing restricted to \mathcal{V} are then given by perfect product trees, for which \text{fbt}_n(T) \in \mathcal{V} for every n \in \omega. The order is given by reverse inclusion. Together we denote them by \mathbb{T}_\mathcal{V}^\otimes.

Let us now take a look at the properties of this forcing notion.

Proposition:  \mathbb{T}_\mathcak{V}^\otimes is proper and {}^\omega\omega-bounding. 

Lemma: Suppose \mathcal{U} is a Ramsey ultrafilter in the ground model and let \mathcal{V} be a P-point in the ground model. Then if:

    \[ \mathtt{0} \Vdash_{\mathbb{T}_\mathcal{V}^\otimes} "\mathcal{U} \text{ does not generate an ultrafilter}"\]

We must have \mathcal{U} \le_{RK} \mathcal{V}. In other words, the Shelah tree forcing destroys only RUF with are below \mathcal{V} in the Rudin-Keisler order of ultrafilters.

Lemma: Let \mathcal{V} be a P-point and \mathbb{T}_\mathcak{V}^\otimes the corresponding Shelah tree forcing. If we have an \omega_2-iteration \P_{\omega_2} of other Shelah tree forcings \mathbb{T}_\mathcak{V_\alpha}^\otimes for some P-points V_\alpha, then they do not intere with each other in the following sense: 

Let G be \mathbb{T}_\mathcak{V}^\otimes-generic over \textbf{V} and G_{\omega_2} be \P_{\omega_2}-generic over \textbf{V}[G], then \mathcal{V} does not generate an ultrafilter in \textbf{V}[G][G_{\omega_2}]. In other words, \mathbb{T}_\mathcak{V}^\otimes killed the ultrafilter \mathcal{V}, and no \omega_2-iterations of Shelah forcing can revive it. 

With these properties, it is possible to prove that for instance the existence of just 27 RUF is consistent with \mathrm{ZFC}. The proof of these properties is rather complicated, and one still needs to prove things such that

  • a RUF will generate a RUF in an extension, if the forcing notion is proper, {}^\omega\omega-bounding, and preserves it as a P-point.
  • non isomorphic RUF stay non isomorphic in {}^\omega\omega-bounding forcing extensions.

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