Subcomplete Forcing, Trees, and Generic Absoluteness

Abstract

We investigate properties of trees of height ω 1 and their preservation under subcomplete forcing. We show that subcomplete forcing cannot add a new branch to an ω 1 -tree. We introduce fragments of subcompleteness which are preserved by subcomplete forcing, and use these in order to show that certain strong forms of rigidity of Suslin trees are preserved by subcomplete forcings. Finally, we explore under what circumstances subcomplete forcing preserves Aronszajn trees of height and width ω 1 . We show that this is the case if CH fails, and if CH holds, then this is the case iff the bounded subcomplete forcing axiom holds. Finally, we explore the relationships between bounded forcing axioms, preservation of Aronszajn trees of height and width ω 1 and generic absoluteness of Σ 1 1 -statements over first order structures of size ω 1 , also for other canonical classes of forcing.

Aronszajn Tree Preservation and Bounded Forcing Axioms

“…Finally, in Section 5, I analyze ( 1 , )-Aronszajn tree preservation under subcomplete forcing systematically, depending on where lies in relation to 2 . For < 2 , the property is provable in ZFC, as was shown in [10]. For = 2 , it is equivalent to the bounded forcing axiom for subcomplete forcing at ( 1 , ).…”

“…Introduction. One of the main observations in Fuchs & Minden [10] was that assuming the continuum hypothesis, the bounded forcing axiom for any natural 1 class Γ of forcing notions that don't add reals is equivalent to the statement that forcing notions in Γ cannot add a cofinal branch to any tree of height and width axiom for any forcing that adds a real fails (since it implies the failure of CH; see [7,Observation 4…”

Subcomplete forcing principles and definable well‐orders

“…The following terminology is from . Definition If is a notion of forcing and is a condition, then is the restriction of to conditions .…”

“…The consistency strength analysis carries over as well, as follows. The version of Fact for ‐closed forcing does not follow from Fact , but instead, one can appeal to [, p. 298, (I6)] and the argument of [, Observation 4.19], under the assumption that . The following theorem can then be proven, using the argument of the proof of Theorem , mutatis mutandis .…”

“…If we slightly abuse notation and denote the resulting principle 2 -BFA <κ-closed , then the version of Observation 4.6 reads: from Fact 3.4, but instead, one can appeal to [22, p. 298, (I6)] and the argument of [13,Observation 4.19], under the assumption that 2 <κ = κ. The following theorem can then be proven, using the argument of the proof of Theorem 4.5, mutatis mutandis.…”