Notes on Improper Forcing

Abstract

In X we prove general theorems on semi-proper forcing notions, and iterations. We use them by iterating several forcing, one of them, and an important one, is Namba forcing. But to show Namba forcing is semi-proper, we need essentially that ℵ ₂ is in a large cardinal which was collapsed to ℵ ₂ (more exactly — a consequence of this on Galvin games). In XI we take great trouble to use a notion considerably more complicated than semi-properness which is satisfied by Namba forcing. However it was not clear whether all this is necessary as we do not exclude the possibility that Namba forcing is always semi-proper, or at least some other forcing, fulfilling the main function of Namba forcing (i.e., making the cofinality of ℵ ₂ to ω without collapsing ℵ ₁). But we prove in 1.2 here, that: there is such semi-proper forcing, iff Namba forcing is semi-proper, iff player II wins Gm({ℵ ₁},ω,ℵ ₂) (a game similar to Galvin games) and, in 1.3, that this implies Chang conjecture. Now it is well known that Chang conjecture implies 0^# exists, so e.g., in ZFC we cannot prove the existence of such semi-proper forcing.