Let be a family of Dense sets in a Forcing Partial Order
Let be a Filter.
Then we say that is -generic if
Furthermore, if is a Transitive Model of some
we say that is -generic over if
and is -generic.
Theorem
If is countable, then there is a -generic Filter.
Proof
Index as
Fix some .
Now construct recursively such that
(possible as is Dense)
Then
is a Filter Base.
The Filter generated by is -generic.
Corollary
If is a countable Transitive Model of some ,
and is a Forcing Partial Order,
then there is a -generic filter over .
Proof
The set is countable.
Therefore, there is a -generic filter (by previous theorem).
This is the -generic filter over .