Aronszajn tree preservation and bounded forcing axioms

Abstract

I investigate the relationships between three hierarchies of reflection principles for a forcing class Γ: the hierarchy of bounded forcing axioms, of Σ 1 1 -absoluteness and of Aronszajn tree preservation principles. The latter principle at level κ says that whenever T is a tree of height ω 1 and width κ that does not have a branch of order type ω 1 , and whenever P is a forcing notion in Γ, then it is not the case that P forces that T has such a branch. Σ 1 1 -absoluteness serves as an intermediary between these principles and the bounded forcing axioms. A special case of the main result is that for forcing classes that don't add reals, the three principles at level 2 ω are equivalent. Special attention is paid to certain subclasses of subcomplete forcing, since these are natural forcing classes that don't add reals.

Canonical fragments of the strong reflection principle

“…Finally, an elementary argument shows that in the situation of Definition 2.21, necessarily, σ↾(2 ω ) N = σ ′ ↾(2 ω ) N , see [21,Obs. 4…”