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.
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Statement
A forcing or partially ordered set P is proper if for all regular uncountable cardinals
The proper forcing axiom asserts that if P is proper and Dα is a dense subset of P for each α<ω1, then there is a filter G
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
Consequences
PFA directly implies its version for ccc forcings, Martin's axiom. In cardinal arithmetic, PFA implies
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
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 ω1. 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
This version of the Proper Iteration Lemma, in which the name
The Proper Iteration Lemma is proved by a fairly straightforward induction on