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Every two elementarily equivalent models have isomorphic ultrapowers

Abstract

We prove (without G.C.H.) that every two elementarily equivalent models have isomorphic ultrapowers, and some related results.

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References

  1. 1.

    C. C. Chang and H. J. Keisler,Model theory, to appear.

  2. 1a.

    Engelking and Karlowicz,Some theorems of set theory and their topological consequences, Fund. Math.57 (1965), 275–285.

    MATH  MathSciNet  Google Scholar 

  3. 2.

    T. Frayne, A. Morel and D. Scott,Reduced direct products, Fund. Math.51 (1962), 195–228.

    MathSciNet  Google Scholar 

  4. 3.

    F. Galvin,Horn sentences, Annals Math. Logic,1 (1970), 389–422.

    MATH  Article  MathSciNet  Google Scholar 

  5. 4.

    J. Los,Quelque remarques, théorèmes et problèmes sur les classes définissable d’algèbres, Mathematical interpretations of formal systems, 98–113, North Holland Publ. Co., Amsterdam, 1955.

    Google Scholar 

  6. 5.

    H. J. Keisler,Reduced products and Horn classes, Trans. Amer. Math. Soc.,117 (1965), 307–328.

    MATH  Article  MathSciNet  Google Scholar 

  7. 6.

    H. J. Keisler,Ultraproducts and elementary classes, Ph. D. thesis, Univ. of Calif. Berkeley California, 1961. See also Kon, Nederl. Akad. Wetensch. Proc. Ser. A64 (1961), 477–495. [Indag. Math. 23].

  8. 7.

    H. J. Keisler,Good ideals in field of sets, Annals of Math.79 (1964), 338–359.

    Article  MathSciNet  Google Scholar 

  9. 8.

    H. J. Keisler,Ultraproducts and saturated models, Koninkl. Nederl. Akad. Wetensch., Proc. Ser. A67 and Indag. Math.26 (1964), 178–186.

    MathSciNet  Google Scholar 

  10. 9.

    H. J. Keisler,A survey of ultraproducts, Proceedings of the 1964 International Congress for logic, methodology and philosophy of science, held in Jerusalem 1964, Bar-Hillel (ed.), North-Holland Publ. Co. Amsterdam, pp. 112–126.

  11. 10.

    H. J. Keisler,Limit ultrapowers, Trans. Amer. Math. Soc.107 (1963), 382–408.

    MATH  Article  MathSciNet  Google Scholar 

  12. 11.

    S. Kochen,Ultraproducts in the theory of models, Annals of Math. (2)74 (1961), 221–261.

    Article  MathSciNet  Google Scholar 

  13. 12.

    K. Kunen,Ultrafilters and independent sets, to appear in Trans. Amer. Math. Soc.

  14. 13.

    R. Mansfield,The theory of Boolean ultrapowers, Annals Math. Logic2 (1971), 297–323.

    MATH  Article  MathSciNet  Google Scholar 

  15. 14.

    R. Mansfield,Horn classes and reduced direct products,a preprint.

  16. 15.

    M. D. Morley and R. L. Vaught,Homogeneous universal models, Math. Scand.11 (1962), 37–57.

    MATH  MathSciNet  Google Scholar 

  17. 16.

    J. L. Bell and A. B. Slomson,Models and ultraproducts, North-Holland Publ. Co. Amsterdam, 1969.

    MATH  Google Scholar 

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The preparation of this paper was supported in Part by NSF Grant GP-22937.

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Shelah, S. Every two elementarily equivalent models have isomorphic ultrapowers. Israel J. Math. 10, 224–233 (1971). https://doi.org/10.1007/BF02771574

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Keywords

  • Equivalent Model
  • Regular Cardinal
  • Elementary Classis
  • Alent Model
  • Proper Filter