Forcing (mathematics) - Countable Transitive Models and Generic Filters

Countable Transitive Models and Generic Filters

The key step in forcing is, given a ZFC universe V, to find appropriate G not in V. The resulting class of all interpretations of P-names will turn out to be a model of ZFC, properly extending the original V (since G∉V).

Instead of working with V, one considers a countable transitive model M with (P,≤,1) ∈ M. By model, we mean a model of set theory, either of all of ZFC, or a model of a large but finite subset of the ZFC axioms, or some variant thereof. Transitivity means that if x ∈ y ∈ M, then x ∈ M. The Mostowski collapsing theorem says this can be assumed if the membership relation is well-founded. The effect of transitivity is that membership and other elementary notions can be handled intuitively. Countability of the model relies on the Löwenheim–Skolem theorem.

Since M is a set, there are sets not in M – this follows from Russell's paradox. The appropriate set G to pick, and adjoin to M, is a generic filter on P. The filter condition means that G⊆P and

• 1 ∈ G ;
• if p ≥ q ∈ G, then p ∈ G ;
• if p,q ∈ G, then ∃r ∈ G, r ≤ p and r ≤ q ;

For G to be generic means

• if D ∈ M is a dense subset of P (that is, p ∈ P implies ∃q ∈ D, q ≤ p) then G∩D ≠ 0 .

The existence of a generic filter G follows from the Rasiowa–Sikorski lemma. In fact, slightly more is true: given a condition p ∈ P, one can find a generic filter G such that p ∈ G. Due to the splitting condition, if G is filter, then P\G is dense. If G is in M then P\G is in M because M is model of set theory. By this reason, generic filter is never in M.