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 \(\aleph _2\).
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Dedicated to the memory of James Baumgartner.
The research presented in this paper was initiated during the Fields Institutes fall 2012 thematic program on Forcing and its Applications; we would like to thank the institute for its generous hospitality. The first author’s research was supported in part by NSF Grant DMS-1262019; the second author’s research was supported in part by Grants from CNRS and NSERC. We would like to thank Garrett Ervin and Hossein Lamei Ramandi for carefully reading earlier version of this paper. We would also like to thank Itay Neeman for pointing out a serious error in an earlier version of this paper.
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Moore, J.T., Todorcevic, S. Baumgartner’s isomorphism problem for \(\aleph _2\)-dense suborders of \(\mathbb {R}\) . Arch. Math. Logic 56, 1105–1114 (2017). https://doi.org/10.1007/s00153-017-0549-4
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DOI: https://doi.org/10.1007/s00153-017-0549-4