# Typical Dynamics of Volume Preserving Homeomorphisms II

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# Forcing axioms and inner models

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Forcing axioms are natural combinatorial statements which decide many

of the questions undecided by the usual axioms ZFC of set theory. The

study of these axioms was initiated in the late 1960s by Martin and

Solovay who introduced Martin's Axiom, followed by the formulation of

the Proper Forcing Axiom by Baumgartener and Shelah in the early 1980s

and Martin's Maximum by Foreman, Magidor and Shelah in the mid-1980s.

In the mid 1990s Woodin's work on Pmax extensions established deep

connections between forcing axioms and the theory of large cardinals

and determinacy. Nevertheless, some of the key problems remained open.

In 2003 Moore formulated the Mapping Reflection Principle (MRP) which

seems to be the missing ingredient needed in order to resolve many of

the remaining open problems in the subject and a number of important

developments followed.

In this lecture I will survey some recent results on forcing axioms:

Moore's work on MRP, my work with A. Caicedo on definable well-orderings

of the reals, Viale's result that the Proper Forcing Axiom implies the

Singular Cardinal Hypothesis, etc.