Iterated ultrapowers for the masses

Open Access


We present a novel, perspicuous framework for building iterated ultrapowers. Furthermore, our framework naturally lends itself to the construction of a certain type of order indiscernibles, here dubbed tight indiscernibles, which are shown to provide smooth proofs of several results in general model theory.


Iterated ultrapower Tight indiscernible sequence Automorphism 

Mathematics Subject Classification

Primary 03C20 Secondary 03C50 


  1. 1.
    Baldwin, J.T.: Categoricity. University Lecture Series 50. American Mathematical Society, Providence (2009)Google Scholar
  2. 2.
    Baldwin, J.T., Larson, P.B.: Iterated elementary embeddings and the model theory of infinitary logic. Ann. Pure Appl. Log. 167, 309–334 (2016)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Boney, W.: Definable Coherent Ultrapowers and Elementary Extensions. arXiv:1609.02970
  4. 4.
    Chang, C.C., Keisler, H.J.: Model Theory. North-Holland, Amsterdam (1973)MATHGoogle Scholar
  5. 5.
    Duby, G.: Automorphisms with only infinite orbits on non-algebraic elements. Arch. Math. Log. 42, 435–447 (2003)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Ehrenfeucht, A., Mostowski, A.: Models of axiomatic theories admitting automorphisms. Fundam. Math. 43, 50–68 (1956)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Enayat, A.: Automorphisms, Mahlo cardinals, and NFU. In: Enayat, A., Kossak, R. (eds.) Nonstandard Models of Arithmetic and Set Theory, Contemporary Mathematics 361, pp. 37–59. American Mathematical Society, Providence (2004)Google Scholar
  8. 8.
    Enayat, A.: Automorphisms of models of bounded arithmetic. Fundam. Math. 192, 37–65 (2006)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Enayat, A.: From bounded arithmetic to second order arithmetic via automorphisms. ASL Lect. Notes Log. 26, 87–113 (2006)MathSciNetMATHGoogle Scholar
  10. 10.
    Enayat, A.: Automorphisms of models of arithmetic: a unified view. Ann. Pure Appl. Log. 145, 16–36 (2007)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Enayat, A., Kaufmann, M., McKenzie, Z.: Largest initial segments pointwise fixed by automorphisms of models of set theory. Arch. Math. Log. doi:10.1007/s00153-017-0582-3
  12. 12.
    Enayat, A., Kaufmann, M., McKenzie, Z.: Appendix to “Iterated ultrapowers of the masses”.
  13. 13.
    Frayne, T., Morel, A.C., Scott, D.S.: Reduced direct products. Fundam. Math. 51, 195–228 (1962/1963)Google Scholar
  14. 14.
    Gaifman, H.: Uniform extension operators for models and their applications. In: Sets, Models and Recursion Theory, pp. 122–155 (Proc. Summer School Math. Logic and Tenth Logic Colloq., Leicester, 1965). North-Holland, Amsterdam (1967)Google Scholar
  15. 15.
    Gaifman, H.: Models and types of Peano’s arithmetic. Ann. Math. Log. 9, 223–306 (1976)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Hodges, W.: Model Theory. Encyclopedia of Mathematics and Its Applications 42. Cambridge University Press, Cambridge (1993)Google Scholar
  17. 17.
    Hrbáček, K.: Internally iterated ultrapowers. In: Nonstandard Models of Arithmetic and Set Theory, Contemporary Mathematics, 361, pp. 87–120. American Mathematical Society, Providence (2004)Google Scholar
  18. 18.
    Jech, T.: Set Theory. Springer, Berlin (2003)MATHGoogle Scholar
  19. 19.
    Kanovei, V., Shelah, S.: A definable nonstandard model of the reals. J. Symb. Log. 69, 159–164 (2004)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Kaufmann, M.: A challenge problem: toward better ACL2 proof technique. In: ACL2 Workshop 2015 Rump Session Abstracts.
  21. 21.
    Kaye, R., Kossak, R., Kotlarski, H.: Automorphisms of recursively saturated models of arithmetic. Ann. Pure Appl. Log. 55, 67–99 (1991)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Kaye, R., Macpherson, D.: Models and groups. In: Kaye, R., Macpherson, D. (eds) Automorphisms of First-Order Structures, pp. 3–31. Oxford University Press, Oxford (1994)Google Scholar
  23. 23.
    Kossak, R., Schmerl, J.H.: The Structure of Models of Arithmetic. Oxford University Press, Oxford (2006)CrossRefMATHGoogle Scholar
  24. 24.
    Körner, F.: Automorphisms moving all non-algebraic points and an application to NF. J. Symb. Log. 63, 815–830 (1998)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Kunen, K.: Some applications of iterated ultrapowers in set theory. Ann. Math. Log. 1, 179–227 (1970)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Łoś, J.: Quelques remarques, theoremes et problemes sur les classes definissables d’algebres. In: Brouwer, L.E.J., Beth, E.W., Heyting, A. (eds.) Mathematical Interpretation of Formal Systems, pp. 98-113. North-Holland, Amsterdam (1955)Google Scholar
  27. 27.
    Marker, D.: Model Theory, Graduate Texts in Mathematics 217. Springer, New York (2002)Google Scholar
  28. 28.
    Poizat, B.: A Course in Model Theory, Universitext. Springer, New York (2000)CrossRefGoogle Scholar
  29. 29.
    Ressayre, J.P.: Review of [G-2]. J. Symb. Log. 48, 484–485 (1983)CrossRefGoogle Scholar
  30. 30.
    Skolem, T.: Über die Nicht-charakterisierbarkeit der Zahlenreihe mittels endlich oder abzä hlbar unendlicht vieler Aussagen mit ausschliesslich Zahlenvariablen. Fundam. Math. 23, 150–161 (1934)CrossRefMATHGoogle Scholar
  31. 31.
    Steel, J.: An introduction to iterated ultrapowers. In: Chong, C., Feng, Q., Slaman, T.A., Woodin, W.H., Yang, Y. (eds.) Forcing, Iterated Ultrapowers, and Turing Degrees. Lecture Notes Series, vol. 29. National University of Singapore, Institute for Mathematical Sciences, Singapore (2015)Google Scholar

Copyright information

© The Author(s) 2017

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Philosophy, Linguistics, and Theory of ScienceUniversity of GothenburgGothenburgSweden
  2. 2.Department of Computer ScienceThe University of Texas at AustinAustinUSA

Personalised recommendations