in Set Theory

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:

  1. 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}.
  2. 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. 
  3. 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:

NotionDominating realsUnbounded realsSplitting 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.

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