In the mathematical field of set theory, the proper forcing axiom (''PFA'') is a significant strengthening of Martin's axiom, where

forcing Forcing may refer to: Mathematics and science * Forcing (mathematics), a technique for obtaining independence proofs for set theory *Forcing (recursion theory), a modification of Paul Cohen's original set theoretic technique of forcing to deal with ...

s with the countable chain condition (ccc) are replaced by proper forcings.

Statement

or partially ordered set P is proper if for all regular uncountable cardinals

\lambda

with P preserves stationary subsets of

\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

\subseteq

P such that D_α ∩ G is nonempty for all α<ω₁. The class of proper forcings, to which PFA can be applied, is rather large. For example, standard arguments show that if P is

ccc CCC may refer to: Arts and entertainment * Canada's Capital Cappies, the Critics and Awards Program in Ottawa, Ontario, Canada * ''Capcom Classics Collection'', a 2005 compilation of arcade games for the PlayStation 2 and Xbox * CCC, the pro ...

or ω-closed, then P is proper. If P is a

countable support iteration In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...

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)

/fin is trivial. PFA implies that the

Singular Cardinals Hypothesis In set theory, the singular cardinals hypothesis (SCH) arose from the question of whether the least cardinal number for which the generalized continuum hypothesis (GCH) might fail could be a singular cardinal. According to Mitchell (1992), the si ...

holds. An especially notable consequence proved by

John R. Steel John Robert Steel (born October 30, 1948) is an American set theory, set theorist at University of California, Berkeley (formerly at University of California, Los Angeles, UCLA). He has made many contributions to the theory of inner models and de ...

is that the axiom of determinacy holds in L(R), the smallest

inner model In set theory, a branch of mathematical logic, an inner model for a theory ''T'' is a substructure of a model ''M'' of a set theory that is both a model for ''T'' and contains all the ordinals of ''M''. Definition Let L = \langle \in \rangle be ...

containing the real numbers. Another consequence is the failure of

square principle In mathematical set theory, a square principle is a combinatorial principle asserting the existence of a cohering sequence of short closed unbounded (club) sets so that no one (long) club set coheres with them all. As such they may be viewed as a ...

s and hence existence of inner models with many

Woodin cardinal In set theory, a Woodin cardinal (named for W. Hugh Woodin) is a cardinal number \lambda such that for all functions :f : \lambda \to \lambda there exists a cardinal \kappa < \lambda with :

\ \subseteq \kappa

and an

Consistency strength

If there is a

supercompact cardinal In set theory, a supercompact cardinal is a type of large cardinal. They display a variety of reflection properties. Formal definition If ''λ'' is any ordinal, ''κ'' is ''λ''-supercompact means that there exists an elementary ...

, 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

\kappa

is supercompact, then there exists a

Laver function In set theory, a Laver function (or Laver diamond, named after its inventor, Richard Laver) is a function connected with supercompact cardinals. Definition If κ is a supercompact cardinal, a Laver function is a function ''ƒ'':κ → ''V ...

for

\kappa

. 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 In set theory, a branch of mathematical logic, Martin's maximum, introduced by and named after Donald Martin, is a generalization of the proper forcing axiom, itself a generalization of Martin's axiom. It represents the broadest class of forcing ...

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 In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least � ...

axioms.

The Fundamental Theorem of Proper Forcing

The Fundamental Theorem of Proper Forcing, due to

Shelah Shelah may refer to: * Shelah (son of Judah), a son of Judah according to the Bible * Shelah (name), a Hebrew personal name * Shlach, the 37th weekly Torah portion (parshah) in the annual Jewish cycle of Torah reading * Salih, a prophet described ...

, states that any

of proper forcings is itself proper. This follows from the Proper Iteration Lemma, which states that whenever

\langle P_\alpha\,\colon\alpha\leq\kappa\rangle

is a countable support forcing iteration based on

\langle Q_\alpha\,\colon
\alpha<\kappa\rangle

and

N

is a countable elementary substructure of

H_\lambda

for a sufficiently large regular cardinal

\lambda

, and

P_\kappa\in N

and

\alpha\in \kappa\cap N

and

p

(N,P_\alpha)

-generic and

p

forces "

q\in P_\kappa/G_\cap N_/math>," then there exists r\in P_\kappa such that r is N -generic and the restriction of r to P_\alpha equals p and p forces the restriction of r to [\alpha,\kappa) to be stronger or equal to q .

This version of the Proper Iteration Lemma, in which the name q is not assumed to be in N, is due to Schlindwein. Schlindwein, C., "Consistency of Suslin's hypothesis, a non-special Aronszajn tree, and GCH", (1994), Journal of Symbolic Logic (59) pp. 1 -- 29 The Proper Iteration Lemma is proved by a fairly straightforward induction on \kappa, and the Fundamental Theorem of Proper Forcing follows by taking \alpha=0 .

References