Adding many Baumgartner clubs

Abstract

I define a homogeneous ℵ 2 -c.c. proper product forcing for adding many clubs of ω 1 with finite conditions. I use this forcing to build models of b(ω 1 ) = ℵ 2 , together with d(ω 1 ) and 2 ℵ 0 large and with very strong failures of club guessing at ω 1 .

A FORCING NOTION COLLAPSING $\aleph _3 $ AND PRESERVING ALL OTHER CARDINALS

“…Let f be the function with domain dom(f) sending α to f(α) ∪ { Q , Q } and let F = F ∪ { Q , Q }. Then (f , F ) is an extension of (f, F ) which is (Q, Add B ( 1 ) L [A] )-generic by Proposition 2.2 (2). Hence, (f , F ) forces the following.…”

“…Only the proof of conclusion (2) is not completely straightforward. For the reader's convenience I am giving a proof of this conclusion suggested by the referee and somewhat simpler than the original proof from [2].…”

“…Add B (X ) is designed to add mutually generic Baumgartner clubs indexed by the ordinals in X while preserving all cardinals. 3 In fact, one obtains Proposition 2.2 by arguments which are either trivial or essentially contained in [2].…”

“…B canonically adds a new club of 1 , which I will call a Baumgartner club. The following definition is from [2]. Definition 2.1.…”

“…(1) In a first step, one proves that there is a cardinal-preserving forcing adding a set A ⊆ 3 such that L[A] computes 3 correctly and such that the collection of internally club 2 elementary submodels of H ( 3 ) L [A] in L[A] is stationary in V. (2) Working in a forcing extension as given by (1), one can extend Abraham's construction, using a natural forcing Add B ( 1 ) for adding ℵ 1 -many mutually generic Baumgartner clubs-rather than Add( , 1 )-in order to predict the relevant objects. Subsequently, Veličković observed that step (2) can be replaced by a significantly simpler argument using Neeman's forcing with chains of models of two types (countable and internally club of size ℵ 1 ) over an extension as given by (1).…”