Proper Forcing Axiom
Encyclopedia
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.
P is proper if for all regular
uncountable cardinals
, forcing
with P preserves stationary subsets
of
.
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
P such that Dα ∩ 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
or ω-closed, then P is proper. If P is a countable support iteration of proper forcings, then P is proper. In general, proper forcings preserve
.
. In cardinal arithmetic
, PFA implies
. PFA implies any two 
-dense subsets of R are isomorphic, any two Aronszajn tree
s are club-isomorphic, and every automorphism of
/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 cardinal
s.
is supercompact, then there exists a Laver function
for
.
It is not yet known how much large cardinal strength comes from PFA.
s 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.
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, the proper forcing axiom (PFA) is a significant strengthening of Martin's axiom
Martin's axiom
In the mathematical field of set theory, Martin's axiom, introduced by , is a statement which is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, so certainly consistent with ZFC, but is also known to be consistent with ZF + ¬ CH...
, where forcings with the countable chain condition
Countable chain condition
In order theory, a partially ordered set X is said to satisfy the countable chain condition, or to be ccc, if every strong antichain in X is countable. There are really two conditions: the upwards and downwards countable chain conditions. These are not equivalent...
(ccc) are replaced by proper forcings.
Statement
A forcing or partially ordered setPartially ordered set
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
P is proper if for all regular
Regular cardinal
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. So, crudely speaking, a regular cardinal is one which cannot be broken into a smaller collection of smaller parts....
uncountable cardinals
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
, forcingForcing (mathematics)
In the mathematical discipline of set theory, forcing is a technique invented by Paul Cohen for proving consistency and independence results. It was first used, in 1963, to prove the independence of the axiom of choice and the continuum hypothesis from Zermelo–Fraenkel set theory...
with P preserves stationary subsets
Stationary set
In mathematics, particularly in set theory and model theory, there are at least three notions of stationary set:-Classical notion:If \kappa \, is a cardinal of uncountable cofinality, S \subseteq \kappa \,, and S \, intersects every club set in \kappa \,, then S \, is called a stationary set....
of
.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
P such that Dα ∩ 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
Countable chain condition
In order theory, a partially ordered set X is said to satisfy the countable chain condition, or to be ccc, if every strong antichain in X is countable. There are really two conditions: the upwards and downwards countable chain conditions. These are not equivalent...
or ω-closed, then P is proper. If P is a countable support iteration of proper forcings, then P is proper. In general, proper forcings preserve
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
.
Consequences
PFA directly implies its version for ccc forcings, Martin's axiomMartin's axiom
In the mathematical field of set theory, Martin's axiom, introduced by , is a statement which is independent of the usual axioms of ZFC set theory. It is implied by the continuum hypothesis, so certainly consistent with ZFC, but is also known to be consistent with ZF + ¬ CH...
. In cardinal arithmetic
Cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality of sets. The cardinality of a finite set is a natural number – the number of elements in the set. The transfinite cardinal numbers describe the sizes of infinite...
, PFA implies
. PFA implies any two -dense subsets of R are isomorphic, any two Aronszajn treeAronszajn tree
In set theory, an Aronszajn tree is an uncountable tree with no uncountable branches and no uncountable levels. For example, every Suslin tree is an Aronszajn tree...
s are club-isomorphic, and every automorphism of
/fin is trivial. PFA implies that the Singular Cardinals Hypothesis holds. An especially notable consequence proved by John R. SteelJohn R. Steel
John Robert Steel is a set theorist at University of California, Berkeley . He has made many contributions to the theory of inner models and determinacy. With Donald A. Martin, he proved projective determinacy, assuming the existence of sufficient large cardinals. He earned his Ph.D...
is that the axiom of determinacy
Axiom of determinacy
The axiom of determinacy is a possible axiom for set theory introduced by Jan Mycielski and Hugo Steinhaus in 1962. It refers to certain two-person games of length ω with perfect information...
holds in L(R)
Constructible universe
In mathematics, the constructible universe , denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis"...
, the smallest inner model
Inner model
In mathematical logic, suppose T is a theory in the languageL = \langle \in \rangleof set theory.If M is a model of L describing a set theory and N is a class of M such that \langle N, \in_M, \ldots \rangle...
containing the real numbers. Another consequence is the failure of square principles and hence existence of inner models with many Woodin cardinal
Woodin cardinal
In set theory, a Woodin cardinal is a cardinal number λ such that for all functionsthere exists a cardinal κ In set theory, a Woodin cardinal is a cardinal number λ such that for all functions...
s.
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 is supercompact, then there exists a Laver functionLaver function
In set theory, a Laver function is a function connected with supercompact cardinals.-Definition:...
for
.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 antichainAntichain
In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two elements in the subset are incomparable. Let S be a partially ordered set...
s of size ω1. Martin's maximum
Martin's maximum
In set theory, Martin's maximum, introduced by , is a generalization of the proper forcing axiom, which is in turn a generalization of Martin's axiom....
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.