Specializing Trees with Small Approximations I

Abstract

We show that under certain appropriate assumptions implied by PFA, every tree of height ω 2 without cofinal branches is specializable via a proper forcing with finite conditions which has the ω 1 -approximation property. The forcing construction employs internally club ω 1 -guessing models as side conditions.

Specialising Trees With Small Approximations II

“…Notice that it may be possible to use Baumgartner's idea [2] to prove Proposition 3.4 with a (possibly) shorter argument, but we believe that the above argument is of independent interest. It is the same idea we used in the main theorem of [13]. Also it resembles the recent analysis of the forcing S ω (T, ω) by Switzer in [15].…”

“…Pick a sufficiently large regular cardinal θ with P(T ) ∈ H θ . The main forcing is defined as in [13]. Let E 0 be the set of <κ-closed and κ-sized elementary submodels M of (H θ , ∈, T ) with M ∩ κ + ∈ κ + .…”

“…As in [13], let us abuse language and say a node t ∈ T is guessed in M ∈ E 0 ∪ E 1 if there is a branch b ∈ M through T such that t ∈ b. Definition 5.3 (P T ).…”

“…In [13], we showed that under a guessing model principle, which follows from PFA, one could generically specialise a tree T of height ω 2 via a proper and ω 2 -preserving forcing with finite conditions. That forcing avoids "countable conditions" in a strong sense as it was shown to have the ω 1 -approximation property.…”

“…In Section 4, we will prepare the ground via Neeman forcing to obtain suitable guessing models and appropriate densely closed ideals together. In Section 5, we sketch how to generalise our previous work [13] using the appropriate circumstances obtained in Section 4.…”