Generic filter

Summary

A generic filter is a filter in a partial order that meets every dense subset belonging to a given model. It is the new object that the method of forcing adds to the model. Because it meets every dense set of the model, it has every property that the model can see as typical. In nontrivial cases it cannot be an element of the model. When the model is countable, generic filters exist by the Rasiowa–Sikorski lemma.