Proper forcing axiom
In the mathematical field of set theory, the proper forcing axiom ("PFA") is a significant strengthening of Martin's axiom, where forcings with the countable chain condition (ccc) are replaced by proper forcings. Statement. A forcing or partially ordered set P is proper if for all regular uncountable cardinals formula_1, forcing with P preserves stationary subsets of formula_2. The proper forcing axiom asserts that if P is proper and D is a dense subset of P for each α<ω, then there is a filter G formula_3 P such that D ∩ G is nonempty for  all α<ω. The class of proper forcings, to which PFA can be applied, is rather large. For example, standard arguments show that if P is ccc or ω-closed, then P is proper. If P is a countable support iteration of proper forcings, then P is proper. Crucially, all proper forcings preserve formula_4. Consequences. PFA directly implies its version for ccc forcings, Martin's axiom. In cardinal arithmetic, PFA implies formula_5. PFA implies any two formula_6-dense subsets of R are isomorphic, any two Aronszajn trees are club-isomorphic, and every automorphism of the Boolean algebra formula_7/fin  is trivial. PFA implies that the Singular Cardinals Hypothesis holds. An especially notable consequence proved by John R. Steel is that the axiom of determinacy holds in L(R), the smallest inner model containing the real numbers. Another consequence is the failure of square principles and hence existence of inner models with many Woodin cardinals. Consistency strength. If there is a supercompact cardinal, then there is a model of set theory in which PFA holds. The proof uses the fact that proper forcings are preserved under countable support iteration, and the fact that if formula_8 is supercompact, then there exists a  Laver function for formula_8. It is not yet known how much large cardinal strength comes from PFA. Other forcing axioms. The bounded proper forcing axiom (BPFA) is a weaker variant of PFA which instead of arbitrary dense subsets applies only to maximal antichains of size ω. Martin's maximum is the strongest possible version of a forcing axiom. Forcing axioms are viable candidates for extending the axioms of set theory as an alternative to large cardinal axioms. The Fundamental Theorem of Proper Forcing. The Fundamental Theorem of Proper Forcing, due to Shelah, states that any countable support iteration of proper  forcings is itself proper. This follows from the Proper Iteration Lemma, which states that whenever formula_10 is a countable support forcing iteration based on formula_11 and formula_12 is a countable elementary substructure of formula_13 for a sufficiently large regular cardinal formula_14, and formula_15 and formula_16 and formula_17 is formula_18-generic and formula_17 forces "formula_20," then there exists formula_21 such that formula_22 is formula_12-generic and the restriction of formula_22 to formula_25 equals formula_17 and formula_17 forces the restriction of formula_22 to formula_29 to be stronger or equal to formula_30. This version of the Proper Iteration Lemma, in which the name  formula_30 is not assumed to be in formula_32, is due to Schlindwein. The Proper Iteration Lemma is proved by a fairly straightforward induction on formula_33, and the Fundamental Theorem of Proper Forcing follows by taking formula_34.