|Authors:||Moore, Justin Tatch
|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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