Definability of the ground model in forcing extensions of ZF-models

Set theory seminarFriday, October 19, 201212:00 amGC 6417

Thomas Johnstone

Definability of the ground model in forcing extensions of ZF-models

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

Richard Laver [2007] showed that if M satisfies ZFC and G is any M-generic filter for forcing P of size less than delta, then M is definable in M[G] from parameter P(delta)^M. I will discuss a generalization of this result for models M that satisfy ZF but only a small fragment of the axiom of choice. This is joint work with Victoria Gitman.

Definition (ZF). P*Q has closure point delta if P is well-orderable of size at most delta and Q is <=delta strategically closed. (Q need not be well-orderable here.) Theorem: If M models ZF+DC_delta and P is forcing with closure point delta, then M is definable in M[G] from parameter P(delta)^M.

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 January 31st, 2013