Consistency
The above can be summarized by saying the fundamental consistency result is that given a forcing poset P, we may assume that there exists a generic filter G, not in the universe V, such that V is again a set theoretic universe, modelling ZFC. Furthermore, all truths in V can be reduced to truths in V regarding the forcing relation.
Both styles, adjoining G to a countable transitive model M or to the whole universe V, are commonly used. Less commonly seen is the approach using the "internal" definition of forcing, and 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.
Read more about this topic: Forcing (mathematics)
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