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Three red herrings around Vaught's conjecture


Authors: John T. Baldwin, Sy D. Friedman, Martin Koerwien and Michael C. Laskowski
Journal: Trans. Amer. Math. Soc. 368 (2016), 3673-3694
MSC (2010): Primary 03C15, 03C55, 03C75, 03E40
DOI: https://doi.org/10.1090/tran/6572
Published electronically: November 6, 2015
MathSciNet review: 3451890
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Abstract: We give a model theoretic proof that if there is a counterexample to Vaught's conjecture there is a counterexample such that every model of cardinality $ \aleph _1$ is maximal (strengthening a result of Hjorth's). In the process we analyze three examples of a sentence characterizing $ \aleph _1$. We also give a new proof of Harrington's theorem that any counterexample to Vaught's conjecture has models in $ \aleph _1$ of arbitrarily high Scott rank below $ \aleph _2$.


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Additional Information

John T. Baldwin
Affiliation: Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, 851 S. Morgan St. (M/C 249), Chicago, Illinois 60607-7045

Sy D. Friedman
Affiliation: Kurt Gödel Research Center for Mathematical Logic, Universität Wien, Währinger Strasse 25, 1090 Wien, Austria

Martin Koerwien
Affiliation: Kurt Gödel Research Center for Mathematical Logic, Universität Wien, Währinger Strasse 25, 1090 Wien, Austria

Michael C. Laskowski
Affiliation: Department of Mathematics, University of Maryland, College Park, Maryland 20742-4015

DOI: https://doi.org/10.1090/tran/6572
Received by editor(s): November 21, 2013
Received by editor(s) in revised form: September 17, 2014
Published electronically: November 6, 2015
Additional Notes: The research of the first author was partially supported by Simons travel grant G5402 and the Austrian Science Fund (FWF)
The research of the second author was supported by FWF (Austrian Science Fund) Grant P24654-N25.
The research of the third author was supported by the Austrian Science Fund (FWF) Lise Meitner Grant M1410-N25
The fourth author was partially supported by NSF grant DMS-1308546
Article copyright: © Copyright 2015 American Mathematical Society