Editing Forcing (mathematics) (section)

== Generic filters ==

Even though each individual forcing condition <math>p</math> cannot fully determine the generic object <math>X</math>, the set <math>G \subseteq \mathbb{P}</math> of all true forcing conditions does determine <math>X</math>. In fact, without loss of generality, <math>G</math> is commonly considered to ''be'' the generic object adjoined to <math>M</math>, so the expanded model is called <math>M[G]</math>. It is usually easy enough to show that the originally desired object <math>X</math> is indeed in the model <math>M[G]</math>.

Under this convention, the concept of "generic object" can be described in a general way. Specifically, the set <math>G</math> should be a '''generic filter''' on <math>\mathbb{P}</math> relative to <math>M</math>. The "[[Filter (mathematics)|filter]]" condition means that it makes sense that <math>G</math> is a set of all true forcing conditions:

* <math> G \subseteq \mathbb{P}; </math>
* <math> \mathbf{1} \in G; </math>
* if <math> p \geq q \in G </math>, then <math> p \in G; </math>
* if <math> p,q \in G </math>, then there exists an <math> r \in G </math> such that <math> r \leq p,q. </math>

For <math> G </math> to be "generic relative to <math>M</math>" means:

* If <math> D \in M </math> is a "dense" subset of <math> \mathbb{P} </math> (that is, for each <math> p \in \mathbb{P} </math>, there exists a <math> q \in D </math> such that <math> q \leq p </math>), then <math> G \cap D \neq \varnothing </math>.

Given that <math>M</math> is a countable model, the existence of a generic filter <math> G </math> follows from the [[Rasiowa–Sikorski lemma]]. In fact, slightly more is true: Given a condition <math> p \in \mathbb{P} </math>, one can find a generic filter <math> G </math> such that <math> p \in G </math>. Due to the splitting condition on <math>\mathbb{P}</math>, if <math> G </math> is a filter, then <math> \mathbb{P} \setminus G </math> is dense. If <math> G \in M </math>, then <math> \mathbb{P} \setminus G \in M </math> because <math> M </math> is a model of <math> \mathsf{ZFC} </math>. For this reason, a generic filter is never in <math> M </math>.