Forcing with Sequences of Models of Two Types

Abstract

Abstract. We present an approach to forcing with finite sequences of models that uses models of two types. This approach builds on earlier work of Friedman and Mitchell on forcing to add clubs in cardinals larger than ℵ 1 , with finite conditions. We use the two-type approach to give a new proof of the consistency of the proper forcing axiom. The new proof uses a finite support forcing, as opposed to the countable support iteration in the standard proof. The distinction is important since a proof using finite supports is more amenable to generalizations to cardinals greater than ℵ 1 . MSC-2010: 03E35.

Forcing Axioms, Finite Conditions and Some More

“…Yet a recent result of Neeman in [25] shows that in some sense we can do exactly this, as he gave an alternative proof of the consistency of PFA using conditions which have finite support. Assuming some bookekepping function f coming from the Laver diamond and giving the list of all proper focings (this is the standard part of any known proof of the consistency of PFA), Neeman introduced the forcing which consists of pairs (M p , w p ) where M p is a finite ∈ −increasing sequence of elementary submodels of H(χ), each either countable or of the form H(α) for some α < χ, and the sequence is closed under intersections.…”

Adding a club with finite conditions, Part II

“…It turns out that these new results drastically simplify the amalgamation results from [3] for strongly adequate sets. As a result we are able to develop a forcing poset for adding a club to a given fat stationary subset of ω 2 with finite conditions which is substantially shorter than our original argument in [3].Forcing posets for adding a club to ω 2 with finite conditions were originally developed by Friedman [1] and Mitchell [5], and then later by Neeman [6]. Adequate set forcing was introduced in [2] in an attempt to simplify and generalize the methods used by the first two authors.…”

“…Forcing posets for adding a club to ω 2 with finite conditions were originally developed by Friedman [1] and Mitchell [5], and then later by Neeman [6]. Adequate set forcing was introduced in [2] in an attempt to simplify and generalize the methods used by the first two authors.…”

Downward Transference of Mice and Universality of Local Core Models

“…However, it was recently shown by Friedman [Fri11] (see also Neeman [Nee14]) that the answer to the PFA question is negative: Using a variant of the method of models as side conditions, Friedman defines a forcing notion that, starting with a supercompact κ, produces an extension where PFA holds and κ becomes ω 2 . The generic for the partial order where only the side conditions are considered also collapses κ to size ω 2 , but the resulting extension does not contain all the reals.…”