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.

A forcing or partially ordered set P is proper if for all regular uncountable cardinals ${\displaystyle \lambda }$, forcing with P preserves stationary subsets of ${\displaystyle [\lambda ]^{\omega }}$.

The proper forcing axiom asserts that if P is proper and D_α is a dense subset of P for each α<ω₁, then there is a filter G ${\displaystyle \subseteq }$ P such that D_α ∩ G is nonempty for all α<ω₁.

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