Three red herrings around Vaught’s conjecture

Baldwin, John; Friedman, Sy; Koerwien, Martin; Laskowski, Michael

doi:10.1090/tran/6572

Three red herrings around Vaught’s conjecture
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by John T. Baldwin, Sy D. Friedman, Martin Koerwien and Michael C. Laskowski PDF

Trans. Amer. Math. Soc. 368 (2016), 3673-3694 Request permission

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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