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Archive for Mathematical Logic

, Volume 56, Issue 7–8, pp 1105–1114 | Cite as

Baumgartner’s isomorphism problem for \(\aleph _2\)-dense suborders of \(\mathbb {R}\)

  • Justin Tatch MooreEmail author
  • Stevo Todorcevic
Article

Abstract

In this paper we will analyze Baumgartner’s problem asking whether it is consistent that \(2^{\aleph _0} \ge \aleph _2\) and every pair of \(\aleph _2\)-dense subsets of \(\mathbb {R}\) 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 \(2^{\aleph _0} \le \aleph _2\) implies that two \(\aleph _2\)-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 \(\aleph _2\) dense suborder X of \(\mathbb {R}\) which cannot be embedded into \(-X\) in any outer model with the same \(\aleph _2\).

Keywords

\(\aleph _2\)-dense Linear order Real type Martin’s Axiom 

Mathematics Subject Classification

03E05 03E35 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsCornell UniversityIthacaUSA
  2. 2.Department of MathematicsUniversity of TorontoTorontoCanada
  3. 3.Institute de Mathématique de JussieuParis CedexFrance

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