Cohen forcing and inner models

Forcing notions in inner models

There is a partial order P preserving stationary subsets of !1 and forcing that every partial order in the ground model V that collapses a su ciently large ordinal to !1 over V also collapses !1 over V P. The proof of this uses a coding of reals into ordinals by proper forcing discovered by Justin Moore and a symmetric extension of the universe in which the Axiom of Choice fails. Also, using on...

Forcing axioms and inner models of set theory

Further Combinatorial Properties of Cohen Forcing

The combinatorial properties of Cohen forcing imply the existence of a countably closed, א2-c.c. forcing notion P which adds a C(ω2)-name Q for a σ-centered poset such that forcing with Q over V P×C(ω2) adds a real not split by V C(ω2) ∩ [ω] and preserves that all subfamilies of size ω1 of the Cohen reals are unbounded.

Inner models with large cardinal features usually obtained by forcing

We construct a variety of inner models exhibiting features usually obtained by forcing over universes with large cardinals. For example, if there is a supercompact cardinal, then there is an inner model with a Laver indestructible supercompact cardinal. If there is a supercompact cardinal, then there is an inner model with a supercompact cardinal κ for which 2κ = κ+, another for which 2κ = κ++ ...

Models of Cohen measurability

We show that in contrast with the Cohen version of Solovay’s model, it is consistent for the continuum to be Cohen-measurable and for every function to be continuous on a non-meagre set.