Authors: Moore, Justin Tatch
Todorčević, Stevo 
Title: Baumgartner’s isomorphism problem for ℵ2 -dense suborders of R
Journal: Archive for Mathematical Logic
Volume: 56
Issue: 7-8
First page: 1105
Last page: 1114
Issue Date: 1-Nov-2017
Rank: M23
ISSN: 0933-5846
DOI: 10.1007/s00153-017-0549-4
Abstract: 
In this paper we will analyze Baumgartner’s problem asking whether it is consistent that 2ℵ0≥ℵ2 and every pair of ℵ2-dense subsets of R are isomorphic as linear orders. The main result is the isolation of a combinatorial principle (∗ ∗) which is immune to c.c.c. forcing and which in the presence of 2ℵ0≤ℵ2 implies that two ℵ2-dense sets of reals can be forced to be isomorphic via a c.c.c. poset. Also, it will be shown that it is relatively consistent with ZFC that there exists an ℵ2 dense suborder X of R which cannot be embedded into - X in any outer model with the same ℵ2.
Keywords: Linear order | Martin’s Axiom | Real type | ℵ -dense 2
Publisher: Springer Link
Project: NSF, Grant DMS-1262019

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