Bagaria’s characterization of bounded forcing axioms in terms of generic absoluteness

Set theory seminarFriday, December 6, 201310:00 am

Thomas Johnstone

Bagaria’s characterization of bounded forcing axioms in terms of generic absoluteness

The New York City College of Technology (CityTech), CUNY

Goldstern and Shelah (1995) introduced the class of bounded forcing axioms, that is forcing axioms for families of antichains of bounded size. For example, the bounded proper forcing axiom ${\rm BPFA}$ asserts that for any proper forcing notion $\mathbb{P}$ and any collection $D$ of at most $\aleph_1$ many maximal antichains in $\mathbb{P}$, each of size at most $\aleph_1$, there is a filter on $\mathbb{P}$ meeting each antichain in $D$. The speaker will present a theorem of Joan Bagaria (2000) that characterizes bounded forcing axioms in terms of generic absoluteness: for instance, Bagaria’s result shows that ${\rm BPFA}$ is equivalent to the assertion that if a $\Sigma_1$ sentence of the language of set theory with parameters of hereditary size at most $\aleph_1$ is true in some proper forcing extension, then it is already true in the ground model.

Thomas Johnstone is a professor of mathematics at the New York City College of Technology (City Tech), CUNY. His research in Set Theory includes large cardinals, forcing, and indestructibility.

Posted by on December 3rd, 2013