Notre Dame Journal of Formal Logic

A Natural Model of the Multiverse Axioms

Victoria Gitman and Joel David Hamkins


If ZFC is consistent, then the collection of countable computably saturated models of ZFC satisfies all of the Multiverse Axioms of Hamkins.

Article information

Notre Dame J. Formal Logic, Volume 51, Number 4 (2010), 475-484.

First available in Project Euclid: 29 September 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03E40: Other aspects of forcing and Boolean-valued models 03E99: None of the above, but in this section

set theory multiverse ZFC forcing


Gitman, Victoria; Hamkins, Joel David. A Natural Model of the Multiverse Axioms. Notre Dame J. Formal Logic 51 (2010), no. 4, 475--484. doi:10.1215/00294527-2010-030.

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  • [1] Barwise, J., and J. Schlipf, "An introduction to recursively saturated and resplendent models", The Journal of Symbolic Logic, vol. 41 (1976), pp. 531--6.
  • [2] Chang, C. C., and H. J. Keisler, Model Theory, 3d edition, vol. 73 of Studies in Logic and the Foundations of Mathematics, North-Holland Publishing Co., Amsterdam, 1990.
  • [3] Friedman, H., "Countable models of set theories", pp. 539--73 in Cambridge Summer School in Mathematical Logic (Cambridge, 1971), vol. 337 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1973.
  • [4] Halimi, B., "Models and universes". preprint.
  • [5] Hamkins, J. D., "The set-theoretical multiverse". submitted.
  • [6] Hamkins, J. D., "The set-theoretical multiverse: A natural context for set theory". submitted.
  • [7] Hamkins, J. D., "Some second order set theory", pp. 36--50 in Logic and Its Applications, edited by R. Ramanujam and S. Sarukkai, vol. 5378 of Lecture Notes in Computer Science, Springer-Verlag, Berlin, 2009.
  • [8] Jensen, D., and A. Ehrenfeucht, "Some problem in elementary arithmetics", Fundamenta Mathematicae, vol. 92 (1976), pp. 223--45.
  • [9] Kaye, R., Models of Peano Arithmetic, vol. 15 of Oxford Logic Guides, The Clarendon Press, New York, 1991.
  • [10] Ressayre, J. P., ``Introduction aux modèles récursivement saturés,'' pp. 53--72 in Séminaire Général de Logique 1983--1984 (Paris, 1983--1984), vol. 27 of PublicationsMathématiques de l'Université Paris VII, Université de Paris VII U.E.R. de Mathématiques, Paris, 1986.
  • [11] Ressayre, J. P., and A. J. Wilkie, Modèles non Standard en Arithmétique et Théorie des Ensembles, vol. 22 of Publications Mathématiques de l'Université Paris VII, Université de Paris VII U.E.R. de Mathématiques, Paris, 1987.
  • [12] Schlipf, J. S., "A guide to the identification of admissible sets above structures", Annals of Pure and Applied Logic, vol. 12 (1977), pp. 151--92.
  • [13] Schlipf, J. S., "Recursively saturated models of set theory", Proceedings of the American Mathematical Society, vol. 80 (1980), pp. 135--42.
  • [14] Scott, D., "Algebras of sets binumerable in complete extensions of arithmetic", pp. 117--21 in Proceedings of Symposia in Pure Mathematics, Vol. V, American Mathematical Society, Providence, 1962.
  • [15] Smoryński, C., "Recursively saturated nonstandard models of arithmetic", The Journal of Symbolic Logic, vol. 46 (1981), pp. 259--86.