Notre Dame Journal of Formal Logic

Transplendent Models: Expansions Omitting a Type

Fredrik Engström and Richard W. Kaye


We expand the notion of resplendency to theories of the kind T+p↑, where T is a first-order theory and p↑ expresses that the type p is omitted; both T and p are in languages extending the base language. We investigate two different formulations and prove necessary and sufficient conditions for countable recursively saturated models of PA.

Article information

Notre Dame J. Formal Logic, Volume 53, Number 3 (2012), 413-428.

First available in Project Euclid: 24 September 2012

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 03C62: Models of arithmetic and set theory [See also 03Hxx]
Secondary: 03C30: Other model constructions

models of arithmetic resplendent models standard cut satisfaction classes


Engström, Fredrik; Kaye, Richard W. Transplendent Models: Expansions Omitting a Type. Notre Dame J. Formal Logic 53 (2012), no. 3, 413--428. doi:10.1215/00294527-1716739.

Export citation


  • [1] Barwise, J., and J. Schlipf, “An introduction to recursively saturated and resplendent models,” Journal of Symbolic Logic, vol. 41 (1976), pp. 531–536.
  • [2] Enayat, A., and A. Visser, “Full satisfaction classes in a general setting, I,” in preparation.
  • [3] Engström, F., “Satisfaction classes in nonstandard models of arithmetic,” Licentiate thesis, Chalmers University of Technology (Sweden), 2002.
  • [4] Engström, F., “Expansions, omitting types, and standard systems,” Ph.D. dissertation, Chalmers University of Technology (Sweden), ProQuest LLC, Ann Arbor, 2004.
  • [5] Hinman, P. G., Recursion-Theoretic Hierarchies, Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1978.
  • [6] Kaye, R., Models of Peano Arithmetic, vol. 15 of Oxford Logic Guides, Clarendon Press, Oxford University Press, New York, 1991.
  • [7] Kaye, R., R. Kossak, and H. Kotlarski, “Automorphisms of recursively saturated models of arithmetic,” Annals of Pure and Applied Logic, vol. 55 (1991), pp. 67–99.
  • [8] Körner, F., “Automorphisms moving all non-algebraic points and an application to NF,” Journal of Symbolic Logic, vol. 63 (1998), pp. 815–830.
  • [9] Kossak, R., and J. H. Schmerl, The Structure of Models of Peano Arithmetic, vol. 50 of Oxford Logic Guides, Clarendon Press, Oxford University Press, Oxford, 2006.
  • [10] Kotlarski, H., S. Krajewski, and A. H. Lachlan, “Construction of satisfaction classes for nonstandard models,” Canadian Mathematical Bulletin, vol. 24 (1981), pp. 283–293.
  • [11] Ressayre, J.-P., “Models with compactness properties relative to an admissible language,” Annals of Mathematical Logic, vol. 11 (1977), pp. 31–55.
  • [12] Smith, S. T., “Extendible sets in Peano arithmetic,” Transactions of the American Mathematical Society, vol. 316 (1989), pp. 337–367.