Proper forcing remastered

Abstract

Abstract. We present the method introduced by Neeman of generalized side conditions with two types of models. We then discuss some applications: a variation of the Friedman-Mitchell poset for adding a club with finite conditions, the consistency of the existence of an ω 2 increasing chain in (ω ω1 1 , < fin ), originally proved by Koszmider, and the existence of a thin very tall superatomic Boolean algebra, originally proved by Baumgartner-Shelah. We expect that the present method will have many more applications.

Forcing with Sequences of Models of Two Types

“…Since then there have already been some applications of the two-type model sequences, for example by Veličković-Venturi [13], using side conditions to obtain new proofs of results of Koszmider, adding a chain of length ω 2 in (ω ω1 1 , < Fin ), and of Baumgartner-Shelah, adding a thin very tall superatomic Boolean algebra. Earlier applications of the Friedman and Mitchell side conditions include Friedman [3], showing that PFA does not imply that a model correct about ℵ 2 must contain all reals, and Mitchell [7], showing that I(ω 2 ) can be trivial.…”

Specializing Trees with Small Approximations I

“…Though the origin goes back to Neeman, we cannot use his results directly here due to the fact we work with different models. Instead, our construction is based on Veličković's presentation [21] of pure side conditions with finite ∈-chains of models of two types in [21], where both types are non-transitive. We will sketch some proofs of the forthcoming facts in this section, but we encourage the reader to consult [21] for more comprehensive proofs.…”

On constructions with 2-cardinals

“…A similar approach focused on the applications of stepping-up in building forcing notions is taken by I. Neeman in [39] or by B. Velickovic and G. Venturi [57] where forcing side conditions have two types of models, those which are countable and those which have cardinality ω 1 .…”