200.239.65.195: /* Further reading */

2014-01-28T20:03:04Z

Further reading

New page

In the mathematical field of [[set theory]], the '''proper forcing axiom''' (''PFA'') is a significant strengthening of [[Martin's axiom]], where [[forcing (set theory)|forcing]]s with the [[countable chain condition]] (ccc) are replaced by proper forcings.

== Statement ==

A [[forcing (set theory)|forcing]] or [[partially ordered set]] P is '''proper''' if for all [[regular cardinal|regular]] uncountable [[cardinal number|cardinals]] <math> \lambda </math>, [[forcing (mathematics) | forcing]] with P preserves [[stationary set|stationary subsets]] of <math> [\lambda]^\omega </math>.

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 <math>\subseteq</math> 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 [[countable chain condition|ccc]] or [[ω-closed]], then P is proper. If P is a [[iterated forcing|countable support iteration]] of proper forcings, then P is proper. In general, proper forcings preserve [[cardinal number|<math>\aleph_1 </math>]].

== Consequences ==
PFA directly implies its version for ccc forcings, [[Martin's axiom]]. In [[Cardinal number|cardinal arithmetic]], PFA implies <math> 2^{\aleph_0} = \aleph_2 </math>. PFA implies any two <math> \aleph_1</math>-dense subsets of R are isomorphic, any two [[Aronszajn tree]]s are club-isomorphic, and every automorphism of <math>P(\omega)</math>/fin is trivial. PFA implies that the [[Singular cardinals hypothesis|Singular Cardinals Hypothesis]] holds. An especially notable consequence proved by [[John R. Steel]] is that the [[axiom of determinacy]] holds in [[Constructible universe|L('''R''')]], the smallest [[inner model]] containing the real numbers. Another consequence is the failure of [[square principle]]s and hence existence of inner models with many [[Woodin cardinal]]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 <math> \kappa </math> is supercompact, then there exists a [[Laver function]] for <math> \kappa </math>.

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 [[antichain]]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.

== References ==

* {{cite book|author=Jech, Thomas|title=Set theory, third millennium edition (revised and expanded)|publisher=Springer|year=2002|isbn=3-540-44085-2|authorlink=Thomas Jech}}
* {{cite journal|author=Steel, John R.|journal=Journal of Symbolic Logic|year=2005|title=PFA implies AD^L(R)|volume=70|issue=4|pages=1255–1296|authorlink=John R. Steel|doi=10.2178/jsl/1129642125}}

[[Category:Axioms of set theory]]

Meshfree methods - Revision history

200.239.65.195: /* Further reading */