Editing Forcing (mathematics) (section)

== Consistency ==

The discussion above can be summarized by the fundamental consistency result that, given a forcing poset <math> \mathbb{P} </math>, we may assume the existence of a generic filter <math> G </math>, not belonging to the universe <math> V </math>, such that <math> V[G] </math> is again a set-theoretic universe that models <math> \mathsf{ZFC} </math>. Furthermore, all truths in <math> V[G] </math> may be reduced to truths in <math> V </math> involving the forcing relation.

Both styles, adjoining <math> G </math> to either a countable transitive model <math> M </math> or the whole universe <math> V </math>, are commonly used. Less commonly seen is the approach using the "internal" definition of forcing, in which no mention of set or class models is made. This was Cohen's original method, and in one elaboration, it becomes the method of Boolean-valued analysis.