Combinatorial properties of classical forcing notions

Combinatorial Properties of Classical Forcing Notions

We investigate the effect of adding a single real (for various forcing notions adding reals) on cardinal invariants associated with the continuum. We show: (1) adding an eventually different or a localization real adjoins a Luzin set of size continuum and a mad family of size ω1; (2) Laver and Mathias forcing collapse the dominating number to ω1, and thus two Laver or Mathias reals added iterat...

[6] J. Brendle, Combinatorial Properties of Classic Forcing Notions, Annals

Combinatorial Properties of Hechler Forcing

Using a notion of rank for Hechler forcing we show: 1) assuming ω 1 = ω L 1 , there is no real in V [d] which is eventually different from the reals in L[d], where d is Hechler over V ; 2) adding one Hechler real makes the invariants on the left-hand side of Cichoń’s diagram equal ω1 and those on the right-hand side equal 2 ω and produces a maximal almost disjoint family of subsets of ω of size...

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.

Simple Forcing Notions and Forcing Axioms