# Blog Archives

# Topic Archive: grounds

# 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.

# Recent progress on the modal logic of forcing and grounds

The modal logic of forcing arises when one considers a model of set theory in the context of all its forcing extensions, with “true in all forcing extensions” and“true in some forcing extension” as the accompanying modal operators. In this modal language one may easily express sweeping general forcing principles, such as the assertion that every possibly necessary statement is necessarily possible, which is valid for forcing, or the assertion that every possibly necessary statement is true, which is the maximality principle, a forcing axiom independent of but equiconsistent with ZFC. Similarly, the dual modal logic of grounds concerns the modalities “true in all ground models” and “true in some ground model”. In this talk, I shall survey the recent progress on the modal logic of forcing and the modal logic of grounds. This is joint work with Benedikt Loewe and George Leibman.