# Blog Archives

# Topic Archive: approximation and cover

# Preservation of DC delta by forcing with a closure point at delta

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.

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

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.