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 $\lambda$, forcing with P preserves stationary subsets of $$lambda\omega.

The proper forcing axiom asserts that if $P$ is proper and $D_\alpha$ is a dense subset of $P$ for each $\alpha < \omega_1$, then there is a filter $G \subseteq P$ such that $D_\alpha \cap G$ is nonempty for all $\alpha < \omega_1$.

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 $\aleph_1$.

Consequences

PFA directly implies its version for ccc forcings, Martin's axiom. In cardinal arithmetic, PFA implies $2^ = \aleph_2$. PFA implies any two $\aleph_1$-dense subsets of R are isomorphic, any two Aronszajn trees are club-isomorphic, and every automorphism of the Boolean algebra $P(\omega)\text$ 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 cardinal

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