Vaught’s conjecture without equality. (English) Zbl 1372.03057

Summary: Suppose that \(\sigma \in \mathcal{L}_{{\omega1},{\omega}}(\mathrm{L})\) is such that all equations occurring in \(\sigma\) are positive, have the same set of variables on each side of the equality symbol, and have at least one function symbol on each side of the equality symbol. We show that \(\sigma\) satisfies Vaught’s conjecture. In particular, this proves Vaught’s conjecture for sentences of \(\mathcal{L}_{{\omega1},{\omega}}(\mathrm{L})\) without equality.


03C15 Model theory of denumerable and separable structures
03C75 Other infinitary logic
03C30 Other model constructions
Full Text: DOI Euclid Link


[1] Barwise, J., Admissible Sets and Structures: An Approach to Definability Theory , vol. 7 of Perspectives in Mathematical Logic , Springer, Berlin, 1975. · Zbl 0316.02047
[2] Hodges, W., Model Theory , vol. 42 of Encyclopedia of Mathematics and its Applications , Cambridge University Press, Cambridge, 1993.
[3] Jech, T., Set Theory , 3rd edition, Springer Monographs in Mathematics , Springer, Berlin, 2003.
[4] Mayer, L. L., “Vaught’s conjecture for o-minimal theories,” Journal of Symbolic Logic , vol. 53 (1988), pp. 146-59. · Zbl 0455.03012 · doi:10.2307/2043741
[5] Morley, M., “The number of countable models,” Journal of Symbolic Logic , vol. 35 (1970), pp. 14-18. · Zbl 0196.01002 · doi:10.2307/2271150
[6] Sági, G., and D. Sziráki, “Some variants of Vaught’s conjecture from the perspective of algebraic logic,” Logic Journal of the IGPL , vol. 20 (2012), pp. 1064-82. · Zbl 1276.03048 · doi:10.1093/jigpal/jzr049
[7] Shelah, S., L. Harrington, and M. Makkai, “A proof of Vaught’s conjecture for \(\omega\)-stable theories,” Israel Journal of Mathematics , vol. 49 (1984), pp. 259-80. · Zbl 0584.03021 · doi:10.1007/BF02760651
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.