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.

Some Properties of Forcing Notions

Suppose $\mathbb{P}$ is a forcing notion in some model $\textbf{V}$ of $\mathrm{ZFC}$ , and let $G$ be a $\mathbb{P}$ -generic filter over $\textbf{V}$ . Then there are a few properties a $\mathbb{P}$ can have that are especially useful.

Adding reals: A lot of forcing notions add reals to the ground model in some form or another. Prominent examples would be Cohen reals for Cohen forcing, Sacks reals for Sacks forcing, etc. However forcing with $\mathbb{P}$ doesn’t only add a single $\mathbb{P}$ -real, as reals definable trough that new real also get added. Hence it would be lovely to have a few properties that describe general reals added by forcing notions. Examples of such properties are:

Adding dominating reals: In every forcing extension $\textbf{V}[G]$ , there is a real $f\in {}^\omega\omega \cap \textbf{V}[G]$ that dominates $^\omega\omega \cap \textbf{V}$ .
Adding unbounded reals: In every forcing extension $\textbf{V}[G]$ , there is a real $f\in {}^\omega\omega \cap \textbf{V}[G]$ that is not bounded by any real in the base model.
Adding splitting reals: In every forcing extension $\textbf{V}[G]$ , there is a real $x \subset \omega$ that splits any real in the base model, i.e.
$\forall y \in \mathcal{P}(\omega) \cap \textbf{V}: |x \cap y| = \omega = |x \setminus y|$

It turns out that if $\mathbb{P}$ adds a dominating real, it also adds both unbounded reals (this is easy) as well as splitting reals (this is not). However none of these inclusions are reversible as shown by the following table, exposing which forcing notions add which types of reals:

Notion	Dominating reals	Unbounded reals	Splitting reals
Mathias	✔	✔	✔
Cohen	✘	✔	✔
Sacks	✘	✘	✘
Silver-like	✘	✘	✔
Miller	✘	✔	✘

Chain conditions: Let $\kappa$ be a regular cardinal. We say that $\mathbb{P}$ satisfies the $\kappa$ -chain condition, if every maximal antichain in $\mathbb{P}$ has size strictly less than $\kappa$ . The case $\kappa = \omega_1$ is also called the countable chain condition or ccc for short. This condition is useful since we have the following

Proposition: If $\mathbb{P}$ satisfies the $\kappa$ -chain condition, then all the cardinals $\lambda \ge \kappa$ are preserved in any forcing extension with $\mathbb{P}$ .

This essentially follows from the fact that if we have a surjective function $f : \alpha \rightarrow \beta$ for some cardinals $\alpha \in \beta$ in the extension, we also have a $\mathbb{P}$ -name $\tilde{f}$ for it. Then for $\lambda \in \alpha$ the set :

$\mathcal{D}_\lambda = \{ p \in \mathbb{P} : \exists \gamma \in \beta \exists q \in \mathbb{P} (q \ge p \wedge q \Vdash_{\mathbb{P}} \tilde{f}(\lambda) = \gamma) \}$

is an antichain in $\mathbb{P}$ , and hence of cardinality strictly smaller than $\kappa$ . Now this set consists of all the conditions above which the value of $f(\lambda)$ will eventually be forced to be some value $\gamma$ . Since $|\mathcal{D}_\lambda| < \kappa$ , we know that number the possible choices for $f(\lambda)$ must also be $< \kappa$ , since every such choice has to be forced by some condition in the filter, by the fundamental theorem of forcing. By the defineability of forcing, we can define another surjective function in the base model:

$g: \bigsqcup_{\lambda \in \alpha} |\mathcal{D}_{\lambda}| \times \alpha \rightarrow \beta$

where the parameter is used to sweep all the possible choices of $\tilde{f}(\lambda)[G]$ . Hence for $\beta > \kappa$ this implies that

$\kappa \times \alpha \ge \bigsqcup_{\lambda \in \alpha} |\mathcal{D}_{\lambda}| \times \alpha \ge \beta > \kappa$

So that $\alpha = \beta$ also in the base model. If $\beta = \kappa$ then if $\alpha<\kappa$ we would have by regularity of $\kappa$ :

$\bigsqcup_{\lambda \in \alpha}|\mathcal{D}_{\lambda}| < \kappa$

A contradiction to the surjectivity of $g$ . Hence in both cases, cardinals $\mu \ge \kappa$ are preserved.

Closedness Properties: Let $\kappa$ be any cardinal. Then $\mathbb{P}$ is called $\kappa$ -closed , if for any ascending sequence of conditions $\langle q_\lambda : \lambda \in \kappa \rangle$ there is a condition $q$ such that for all $\lambda \in \kappa : q \ge q_\lambda$ . For $\kappa = \omega$ , this is called $\sigma$ -closedness. This property complements the $\kappa$ -chain condition, in that the following hols:

Proposition: If $\mathbb{P}$ satisfies the $\kappa$ -closedness condition, then all the cardinals $\lambda \le \kappa$ are preserved in any forcing extension with $\mathbb{P}$ .

This follows from the fact, that if $f: \lambda \rightarrow X$ is a function in $\textbf{V}[G]$ with $\lambda \le \kappa$ , then $f$ is already in the ground-model. Hence no cardinals less than $\kappa$ can be collapsed. Now why is this the case? $f$ has a $\mathbb{P}$ -name $\tilde{f}$ in $\mathbf{V}$ . Start with any condition $p$ . Define the condition $q_0\ge p$ by:

$q_0 \Vdash_{\mathbb{P}} \tilde{f}(0) = x_0$

for some $x_0\in X$ . This is possible, since the conditions that define $\tilde{f}(0)[G]$ are open dense in $P$ . Again by open density find $q_1 \ge q_0$ such that for some $x_1 \in X$ with:

$q_1 \Vdash_{\mathbb{P}} \tilde{f}(1) = x_1$

Continue in this manner to obtain an ascending sequence $q_0 \le q_1 \le q_2 \le \cdots$ . Now at the limit stage, utilize the $\kappa$ -closedness to find $q_\omega \ge q_k$ for any $k \in \omega$ . Continue this process further until we have a condition $q_\lambda$ such that for all $\alpha \in \lambda$ :

$q_\lambda \Vdash_{\mathbb{P}} \tilde{f}(\alpha) = x_\alpha$

So above each $p$ , there is a $q_\lambda$ which essentially says that $f$ is definable in $\textbf{V}$ , hence no matter which generic filter we choose, there will be some $q_\lambda \in G$ , and hence $f$ is already a function in $\textbf{V}$ .

Properness: A generalization of the $ccc$ . Remember that $ccc$ assures that no uncountable cardinals get collapsed. Also $ccc$ gets preserved under finite support iterations. However this is often too much to ask from a forcing notion, hence the notion of properness, for which the following hold:

Proposition 1: If $\mathbb{P}$ is proper, then it does not collapse $\omega_1$ .

Proposition 2: If $\mathbb{P}_\alpha$ is a countable support iteration of $\langle \tilde{\mathbb{Q}_\beta : \beta \in \alpha}\rangle$ , and for each $\beta \in \alpha$ we have $\mathtt{0}_\beta \Vdash_{\mathbb{P}} "\tilde{\mathbb{Q}}_\beta \text{ is proper }"$ , then $\mathbb{P}_\alpha$ is proper.

Proposition 3: If $\mathbb{P}\alpha$ is a countable support iteration of proper forcing notions which satisfy one of the following properties

Laver Property
Preservation of P-points
${}^\omega\omega$ -boundedness

Then $\mathbb{P}\alpha$ is proper and has the respective property.

So now to the definition of properness. For a regular limit cardinal $\chi$ let:

$\textbf{H}_\chi = \{ x \in \textbf{V}_\chi : |\text{TC}(x)| < \chi\}$

These are all the sets which are hereditarily of cardinality $< \chi$ . It turns out $\textbf{H}_\chi$ is a model of $\mathrm{ZFC}$ minus the power set axiom. For the definition of properness, we need $\chi$ large enough. Assume $\chi > \beth^+_\omega$ in the following, this will suffice. Let $\textbf{N} = (N, \in)$ be an elementary submodel of $(\textbf{H}_\chi, \in)$ , i.e. any statement about elements of $N$ is true in $(\textbf{H}_\chi, \in)$ iff the corresponding statement (where all the quantifiers have to be relativised) is true in $textbf{N}$ .

Let $(P, \le) \in N$ . We say that $G \subset P$ is $\textbf{N}$ -generic if it has the following property: If $D \in N$ and $\textbf{N} \vDash "G \text{ is an open dense subset of P}"$ . So if the model $\textbf{N}$ says that a subset $D \subset P$ is open dense, $G$ should intersect it, similarly to genericity over $\textbf{V}$ . Next, a condition $q \in P$ , which does not need to be in $N$ , is $\textbf{N}$ -generic if

$\textbf{V} \vDash q \Vdash_{\mathbb{P}} "\dot{G} \text{ is } \textbf{N}\text{-generic}"$

where $\dot{G}$ is the canonical $\mathbb{P}$ -name for the $\mathbb{P}$ -generic filter in $\textbf{V}$ .

Finally, a forcing notion $\mathbb{P}$ is called proper, if for all countable elementary submodels $\textbf{N}$ of $(\texbf{H}_\chi, \in)$ which contain $\mathbb{P}$ , and every condition $p \in N \cap P$ , there is a $\textbf{N}$ -generic condition $q \ge p$ .

Laver Property: A property that makes sure no Cohen reals are added while forcing. Let $\mathcal{F}$ be the family of all function $s: \omega \rightarrow \text{fin}(\omega)$ , such that $\forall n\in \omega: |s(n)| \le 2^n$ . The family $\mathcal{F}$ is restricting the growth of its member to exponential growth. A forcing notion $\mathbb{P}$ is said to have the Laver property, if for any function $f\in {}^\omega\omega$ in the ground model and any $\mathbb{P}$ -name $\tilde{g}$ of a function $g \in {}^\omega\omega \cap \textbf{V}[G]$ , such that $\mathtt{0} \Vdash_{\mathbb{P}} \forall n \in \omega: (\tilde{g}(n) < \dot{f}(n))$ we have:

$\mathtt{0} \Vdash_{\mathbb{P}} \exists s \in \dot{\mathcal{F}}\forall n \in \omega (\tilde{g}(n) \in s(n))$

Another way of looking at the Laver property is that if $\mathbb{P}$ satisfies the Laver property, the models $\textbf{V}$ and $\textbf{V}[G]$ cannot be too wildly different, since each bounded funtion in the extended model can at least be approximated in the ground model.

Preserving P-points: Let $\mathcal{U} \subset [\omega]^\omega$ be a P-point in $\textbf{V}$ . Then we say that $\mathbb{P}$ preserves $\mathcal{U}$ , if $\dot{\mathcal{U}}[G]$ generates an ultrafilter in any extension. Equivalently:

$\mathtt {0} \Vdash_{\mathbb{P}} "\mathcal{U} \text{ generates an ultrafilter }"$

Note, that we don’t require the ultrafilter generated in the extension to be a P-point. This is where the following statement comes in:

Proposition: Let $\mathbb{P}$ be a proper forcing notion and $\mathcal{U}$ be a P-point in the ground model $\textbf{V}$ . If the filter generated in $\textbf{V}[G]$ is an ultrafilter, then it is a P-point.

So nice enough forcing notions maintain P-points in this stronger sense.

Furthermore, the countable support iterations of P-point preserving proper forcing notions also preserves P-points.

${}^\omega\omega$ -bounding: A forcing notion $\mathbb{P}$ is called ${}^\omega\omega$ -bounding if there are no unbounded reals in any forcing extension.

${}^\omega\omega$ -boundedness of proper forcing notions is preserved under countable support iterations.

Some Forcing Notions

In this post I will give a few examples of forcing notions and explore their combinatorial properties. Most of this is directly taken from “combinatorial set theory with a gentle introduction to forcing” by Halbeisen and condensed down a bit.

Ultrafilter Forcing $\mathbb{U} := ([\omega]^\omega, \supseteq^*)$ :

What makes this one useful, is that a $\mathbb{U}$ -generic filter $G$ over $\textbf{V}$ is also a Ramsey Ultrafilter in the extension, different from all ultrafilters in the ground model. Hence, when forcing with this notion, one adds at least one Ramsey Ultrafilter to the ground model. There are a few things one needs to check to be able to conclude:

RUF’s $\math$ are characterised by the fact that for each coloring

$\pi : [\omega]^2 \rightarrow 2$

there is a $x \in \mathcal{U}$ such that $\pi|_{[x]^2}$ is constant, i.e. for each coloring, $\mathcal{U}$ contains a monochromatic set. We will conclude that the new filter $G$ has such a set for each coloring in the ground model. However one also needs to make sure that no new colorings get added, so that indeed the property holds true for all colorings in $\textbf{V}[G]$ . For this one checks that $\mathbb{U}$ is $\omega_1$ -closed, hence preserves all cardinals $\le \omega_1$ , and in particular all real numbers $r : \omega \rightarrow \omega$ . Since the reals are in one-to one correspondence with the two colorings of $[\omega]^2$ , there are no new colorings appearing in $\textbf{V}[G]$ .
$G$ is in fact an ultrafilter which is different from every ultrafilter in the ground model. For the first fact consider the open dense set:
$\mathcal{D}_x = \{ y \in [\omega]^\omega : x \supseteq^* y \vee \omega \setminus x \supseteq^* y\}$

$G$ being $\mathbb{U}$ -generic means it intersects this set, and hence either $x \in G$ or $\omega \setminus x \in G$ . For the second fact let $\mathcal{V}$ be an ultrafilter in the ground model and consider the following open dense set:
$\mathcal{D}_{\mathcal{V}} = \{ y \in [\omega]^\omega : y \not\in \mathcal{V}\}$
Here, $G$ being $\mathbb{U}$ -generic implies, that for every ultrafilter $\mathcal{V}$ in the ground model, there is a set in $G$ that is not contained in $\mathcal{V}$ , hence in particular $\mathcal{V} \ne G$ .

Cohen Forcing $\mathbb{C}_\kappa = (\text{Fn}(\omega \times \kappa, 2), \subset)$ :

This forcing notion can be used to show that $\neg \textbf{CH}$ is consistent with $\textbf{ZFC}$ . Suppose $\textbf{V}$ was a model of $\textbf{ZFC}+\neg \textbf{CH}$ . Then forcing with $\mathbb{C}_\kappa$ adds $\kappa$ distinct new reals to the model, the so called Cohen reals, and preserves all cardinals. In fact if $G$ is $\mathbb{C}_\kappa$ -generic over $\textbf{V}$ , we get that $\bigcup G$ is a function $\omega \times \kappa \rightarrow 2$ . This can be concluded by looking at the open dense set:

$\mathcal{D}_\lambda = \{ p \in \text{Fn}(\omega \times \kappa, 2) : \alpha \in \text{dom}(p)\}$

which shows that $\bigcup G$ is defined for all $\lambda \in \kappa$ . Since $\mathfrak{c}$ counts exactly the number of reals, if we choose $\kappa > \omega_1$ we get:

$\mathfrac{c} \ge \kappa > \omega_1 .$

Hence we see that the continuum hypothesis no longer holds. However there are a few things one needs to check given a $\mathbb{C}_\kappa$ -generic filter $G$ over $\textbf{V}$ :

The added reals are indeed are all distinct and different from all existing reals. First, for $\lambda \in \kappa$ , we define the new reals by:
$r_\lambda = \left( \bigcup G \right)|_{\omega \times \{ \lambda\}}$

Then you look at:
$\mathcal{D}_{\lambda, \lambda'} = \{ p \in \text{Fn}(\omega \times \kappa, 2): \exists n \in \omega (p(\lambda, n) \ne p(\lambda', n))\}$

which exactly tells you that no two $r_\lambda$ ‘s are the same, and consider a similar open dense set to show that they also differ from all reals in the ground model.
The forcing doesn’t collapse any cardinals, since otherwise $\kappa$ might shrink down to $\omega_1$ again and we wouldn’t have proved anything. This is shown using the fact that $\mathbb{C}_\kappa$ is $ccc$ by a consequence of the $\Delta$ -system lemma, and hence preserves all infinite cardinals.

Collapsing Forcing $\mathbb{K}_{\kappa, \omega_1}$ :

The conditions are given by partial functions $\omega_1 \rightarrow \kappa$ with countable domain, and the order is the inclusion. As the name suggests, this forcing can be used to collapse a cardinal $\kappa$ to $\omega_1$ . In particular for $\kappa = \mathfrak{c}$ we have that for a $\mathbb{K}_{\kappa, \omega_1}$ -generic filter over $\mathbf{V}$ :

$\textbf{V}[G] \vDash \textbf{CH}$

One needs to show a couple of things for that:

No new reals get added. This can be assured by verifying the $\sigma$ -closedness of $\mathbb{K}_{\kappa, \omega_1}$ .
$\mathbb{K}_{\kappa, \omega_1}$ adds a surjective function $f : \omega_1 \twoheadrightarrow \kappa$ , so that
$\textbf{V}[G] \vDash \kappa \le \omega_1$

Sacks Forcing $\mathbb{S}$ :

Here the conditions are given by:

$S = \{ T \subset {}^{<\omega}2 : T \text{ is a perfect tree}\}$

and $\mathbb{S} = (S, \supset)$ . A tree $T \subset {}^{<\omega}2 = \bigcup_{n \in \omega} {}^n2$ is a subset that is closed under taking initial segments, i.e. if $p \in T$ , then $p|_k \in T$ for all $k \in \omega$ . We also say that $t$ extends $s$ , written $s \prec t$ , if $s$ is an initial segment of $t$ . A perfect tree, is a tree that splits indefinitely, i.e. $\forall t \in T \exists t' \ne t''\in T: t \prec t' \wedge t \prec t''$ . Sacks forcing allows us to build a model $\textbf{V}$ [G], such that there is no “intermediate model” between $\textbf{V}$ and $\textbf{V}$ [G], meaning if $\textbf{V} \subset \textbf{W} \subset \textbf{V}[G]$ is a model of $\mathrm{ZFC}$ then either $\textbf{V} = \textbf{W}$ or $\textbf{W} = \textbf{V}[G]$ . Forcing with perfect trees was used by Gerald Enoch Sacks to produce a real a with minimal degree of constructibility.

Sacks Forcing adds neither splitting nor unbounded reals

Miller Forcing $\mathbb{M}$ :

Here the conditions are given by:

$M = \{ T \subset \text{seq}(\omega) : T \text{ is a superperfect tree} \}$

and $\mathbb{M} = (M, \supset)$ . Here superperfect means that for $s \in T$ :

$\text{next}(t) := \{ t\in T: \text{dom}(t) = \text{dom}(s)+1 \wedge s \prec t\}$

is infinite for all $t \in T$ .

It adds unbounded reals but no splitting reals.

Mathias Forcing $\mathbb{M}_{\mathcal{U}}$ :

where $\mathcal{U}$ is a fixed Ramsey family. The conditions are given by:

$M_{\mathcal{U}} = \{ (s,x) : s\in \text{fin}(\omega) \land x \in \mathcal{U}   \land \text{max}(s) < \text{min}(x)\}$

and the preorder is given by:

$(s,x) \le (t, y) \Leftrightarrow s \subset t \land y \subset x \land t \setminus s \subset x$

Mathias forcing can be used to construct a pseudointersection of an ultrafilter.

Restricted Laver Forcing $\mathbb{L}_{\mathcal{U}}$ :

Here $\mathcal{U} \subset [\omega]^\omega$ is a fixed ultrafilter. Now a Laver tree is a tree $T \subset \text{seq}(\omega)$ if it has a special element $s \in T$ , called the stem, such that for any $t \in T$ we have $t \prec s \vee s \prec t$ , and for every $t \in T$ with $s \prec t$ , we have that $\text{next}(t)$ is infinite. A Laver tree restricted to $\mathcal{U}$ is a Laver tree such that $\text{next}(t) \in \mathcal{U}$ . The conditions of Laver forcing are exactly the Laver trees restricted to $\mathcal{U}$ , with reverse inclusion.

This forcing is $\sigma$ -centred and adds a real which is almost homogenous for all colorings in the ground model.

Silver Forcing $\mathbb{S}_{\mathcal{E}}$ :

Here $\mathcal{E}$ is a fixed P-family. Define:

$S_\mathcal{E} = \bigcup\{ {}^x2: \omega \setminus x \in \mathcal{E}\}$

and for $p,q \in S_\mathcal{E}$ :

$p \le q \Leftrightarrow \text{dom}(p) \subset \text{dom}(q) \wedge q|_{\text{dom}(p)} = p$

Then $\mathbb{S}_{\mathcal{E}} = (S_\mathcal{E}, \le)$ . For $\mathcal{E} = [\omega]^\omega$ write $\mathbb{S}$ and just call it Silver forcing. Silver reals are given by:

$r = \bigcup G$

In general, Silver filters cannot be reconstructed from Silver reals, since multiple Silver filters yield the same real. However, if $\mathcal{E}$ is an ultrafilter we can always do the reconstruction.

Shelah’s Tree forcing $\mathbb{T}$ : See my other post here.