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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