# 2008 seminar talk: The consistency b = kappa ^lt; s = kappa^+

Talk held by Vera Fischer (KGRC) at the KGRC seminar on 2008-10-09.

### Abstract

In 1984 S. Shelah obtains the consistency of b = omega_1 < s = omega_2 using a proper forcing notion of size continuum, which adds a real not split by the ground model reals and satisfies the almost omega^omega-bounding property. We obtain a sigma-centered suborder of Shelah's poset, which behaves very similarly to the larger forcing notion: it adds a real not split by the ground model reals and preserves the unboundedness of a chosen unbounded, directed family of reals. Thus under an appropriate finite support iteration of length kappa^+, where kappa is an arbitrary regular uncountable cardinal, we obtain the consistency of b = kappa < s = kappa^+