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On the definability of the quantifier “there exist uncountably many”

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Abstract

In paper [5] it was shown that a great part of model theory of logic with the generalized quantifier Q x = “there exist uncountably many x” is reducible to the model theory of first order logic with an extra binary relation symbol. In this paper we consider when the quantifier Q x can be “syntactically” defined in a first order theory T. That problem was raised by Kosta Došen when he asked if the quantifier Q x can be eliminated in Peano arithmetic. We answer that question fully in this paper.

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References

  1. H. R. Jervel, Conservative endextensions and the quantifier “there exist uncountably many”. In: Proceedings of the Third Scandinavian Logic Symposium, Uppsala, 1973 (ed. S. Kanger), North-Holland Publishing Company, 1975, pp. 63–80.

  2. H. J. Keisler, Logic with quantifier “there exist uncountably many”, Annals of Mathematical Logic, 1 (1970), pp. 1–93.

    Google Scholar 

  3. H. J. Keisler, Some model-theoretic results for ω-logic, Israel Journal of Mathematics, 4 (1966), pp. 249–261.

    Google Scholar 

  4. Ž. Mijajlović, A note on elementary endextension, Publications de l'Institut Mathématique, (Belgrade), 21 (1977), pp. 141–144.

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  5. Ž. Mijajlović and V. Harizanov, Regular relations and the quantifier “there exist uncountably many”, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, Bd. 29 (1983), pp. 151–161.

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  6. M. Morley and R. Vaught, Homogeneous universal models, Mathematica Scandinavica, 11 (1962), pp. 37–57.

    Google Scholar 

  7. J. S. Shoenfield, Mathematical Logic, Addison-Wesley Publishing Company Reading, Mass. (1973).

    Google Scholar 

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I would like to thank Kosta Došen and Zoran Marković who made valuable suggestions and remarks on a draft of this paper.

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Mijajlović, Ž. On the definability of the quantifier “there exist uncountably many”. Stud Logica 44, 257–264 (1985). https://doi.org/10.1007/BF00394445

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  • DOI: https://doi.org/10.1007/BF00394445

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