A dichotomy for forcing notions

Simple Forcing Notions and Forcing Axioms

On Nicely Definable Forcing Notions

We prove that if Q is a nw-nep forcing then it cannot add a dominating real. We also show that amoeba forcing cannot be P(X)/I if I is an א1-complete ideal. Furthermore, we generalize the results of [12].

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...

Universal forcing notions and ideals

The main result of this paper is a partial answer to [6, Problem 5.5]: a finite iteration of Universal Meager forcing notions adds generic filters for many forcing notions determined by universality parameters. We also give some results concerning cardinal characteristics of the σ–ideals determined by those universality parameters. One of the most striking differences between measure and catego...

More forcing notions imply diamond

We prove that the Sacks forcing collapses the continuum onto d, answering the question of Carlson and Laver. Next we prove that if a proper forcing of the size at most continuum collapses ω2 then it forces ♦ω1 . Research partially supported by KBN 654/2/91 Research partially supported by “Basic Research Foundation” administered by The Israel Academy of Sciences and Humanities. Publication 475.