Andrews, Uri; Goldbring, Isaac; Hachtman, Sherwood; Keisler, H. Jerome; Marker, David Scattered sentences have few separable randomizations. (English) Zbl 1481.03013 Arch. Math. Logic 59, No. 5-6, 743-754 (2020). Summary: In the paper [in: Beyond first order model theory. Boca Raton, FL: CRC Press. 189–220 (2017; Zbl 1433.03097)], H. J. Keisler showed that if Martin’s axiom for aleph one holds, then every scattered sentence has few separable randomizations, and asked whether the conclusion could be proved in ZFC alone. We show here that the answer is “yes”. It follows that the absolute Vaught conjecture holds if and only if every \(L_{\omega_1\omega } \)-sentence with few separable randomizations has countably many countable models. MSC: 03C15 Model theory of denumerable and separable structures 03C65 Models of other mathematical theories 03C90 Nonclassical models (Boolean-valued, sheaf, etc.) Keywords:randomizations; scattered sentences; Vaught’s conjecture Citations:Zbl 1433.03097 PDFBibTeX XMLCite \textit{U. Andrews} et al., Arch. Math. Logic 59, No. 5--6, 743--754 (2020; Zbl 1481.03013) Full Text: DOI arXiv References: [1] Andrews, U.; Keisler, HJ, Separable models of randomizations, J. Symb. Log., 80, 1149-1181 (2015) · Zbl 1373.03051 · doi:10.1017/jsl.2015.33 [2] Baldwin, J.; Friedman, S.; Koerwien, M.; Laskowski, M., Three red herrings around Vaught’s conjecture, Trans. Am. Math. Soc., 368, 3673-3694 (2016) · Zbl 1528.03154 · doi:10.1090/tran/6572 [3] Ben Yaacov, I., Berenstein, A., Henson, C.W., Usvyatsov, A.: Model theory for metric structures. In: Model Theory with Applications to Algebra and Analysis. London Mathematical Society Lecture Note Series, vols. 2, 350, pp. 315-427 (2008) · Zbl 1233.03045 [4] Eagle, C.; Vignati, A., Saturation and elementary equivalence of \(C^*\)-algebras, J. Funct. Anal., 269, 2631-2664 (2015) · Zbl 1477.46057 · doi:10.1016/j.jfa.2015.04.013 [5] Jech, T., Set Theory (2003), New York: Springer, New York · Zbl 1007.03002 [6] Keisler, HJ, Model Theory for Infinitary Logic (1971), Amsterdam: North-Holland, Amsterdam · Zbl 0222.02064 [7] Keisler, HJ; Iovino, J., Randomizations of scattered sentences, Beyond First Order Model Theory, 189-220 (2000), Boca Raton: CRC Press, Boca Raton · Zbl 1433.03097 [8] Morley, M., The number of countable models, J. Symb. Log., 35, 14-18 (1970) · Zbl 0196.01002 · doi:10.1017/S0022481200092173 [9] Scott, D.; Addison, J., Logic with denumerably long formulas and finite strings of quantifiers, The Theory of Models, 329-341 (1965), Amsterdam: North-Holland, Amsterdam · Zbl 0166.26003 [10] Steel, JR; Kechris, AS; Moschovakis, YN, On Vaught’s conjecture, Cabal Seminar 76-77. Lecture Notes in Mathematics, 193-208 (1978), New York: Springer, New York · Zbl 0403.03027 [11] Vaught, R.L.: Denumerable models of complete theories. In: Infinitistic Methods, Warsaw, pp. 303-321 (1961) · Zbl 0113.24302 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.