Finite Function Forcing

Let and be sets and define

i.e. is a finite Partial Function from to . Furthermore, define

and set . Then is a Forcing Partial Order.

Lemma

For , they are Incompatible if and only if

Lemma

If is a Filter Base then

is a function. Furthermore, if is a Filter then is a function.

Lemma

For any define:

and for define:

Then is Dense, and is Dense when is infinite. Furthermore, if is a -Generic Filter then

and if is a -Generic Filter and is infinite then

Lemma

Let be infinite. Fix and define

Then is Dense. Furthermore, if is a -Generic Filter then