Adding a club with finite conditions, Part II

Abstract

Abstract. We define a forcing poset which adds a club subset of a given fat stationary set S ⊆ ω 2 with finite conditions, using S-adequate sets of models as side conditions. This construction, together with the general amalgamation results concerning S-adequate sets on which it is based, is substantially shorter and simpler than our original version in [3].The theory of adequate sets introduced in [2] provides a framework for adding generic objects on ω 2 with finite conditions using countable models as side conditions. Roughly speaking, an adequate set is a set of models A such that for all M and N in A, M and N are either equal or membership comparable below their comparison point β M,N . A technique which was central to the development of adequate sets in [2], as well as to our original forcing for adding a club to a fat stationary subset of ω 2 in [3], involves taking an adequate set A and enlarging it to an adequate set which contains certain initial segments of models in A.In this paper we prove amalgamation results for adequate sets which avoid the method of adding initial segments of models. 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. This new framework is also flexible as it admits useful variations. For example, in a subsequent paper [4] we show that the forcing poset for adding a club presented below can be modified to preserve CH, answering a problem of Friedman [1].

Absoluteness via resurrection

“…The results of this paper outline that any interesting iteration theorem for a class Γ ⊇ Γ ω2 closed under two step iterations, can be used to prove that RA ON (Γ) is consistent relative to suitable large cardinal assumptions and that it freezes the theory of H ω3 with respect to forcings in Γ preserving RA ω (Γ). It is nonetheless still a mystery which classes Γ ⊇ Γ ω2 can give us a nice iteration theorem, even if the recent works, by Neeman, Asperò, Krueger, Mota, Velickovic and others are starting to shed some light on this problem (see among others [20,21,25]).…”

Forcing with adequate sets of models as side conditions

“…Since these three kinds of objects can be forced using classical methods, the purpose of these examples is to illustrate the method, rather than proving new consistency results. This is the first in a series of papers which develop the adequate set approach to forcing with side conditions on ω 2 ( [7], [6], [8], [9]). While many of the arguments appearing here could, with some work, be subsumed in the previous frameworks of Friedman, Mitchell, and Neeman, this paper is important for presenting the basic ideas of adequate sets in a way which provides a foundation for further developments.…”

Mitchell's theorem revisited

“…(4) is similar. In this section we will provide a detailed analysis of remainder points; some of these arguments appeared previously in [8] and [9], although in a less complete form. This analysis will be the foundation from which we derive the amalgamation results of Section 13.…”

“…Notably, this approach has led to the solution of an open problem of Friedman [3], by showing how to add a club subset of ω 2 with finite conditions while preserving the continuum hypothesis ( [10]). Other applications are given in [8], [7], [9], and [2].…”

“…More generally, it is possible to add a club subset to a fat stationary subset of ω 2 using adequate sets as side conditions, but the argument is more complicated than the one which we give here. See[9].…”