Notre Dame Journal of Formal Logic

Definability of Initial Segments

Saharon Shelah and Akito Tsuboi

Abstract

In any nonstandard model of Peano arithmetic, the standard part is not first-order definable. But we show that in some model the standard part is definable as the unique solution of a formula $\varphi(P)$, where P is a unary predicate variable.

Article information

Source
Notre Dame J. Formal Logic, Volume 43, Number 2 (2002), 65-73.

Dates
First available in Project Euclid: 15 December 2003

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1071509428

Digital Object Identifier
doi:10.1305/ndjfl/1071509428

Mathematical Reviews number (MathSciNet)
MR2033316

Zentralblatt MATH identifier
1082.03038

Subjects
Primary: 03C62: Models of arithmetic and set theory [See also 03Hxx] 03H15: Nonstandard models of arithmetic [See also 11U10, 12L15, 13L05]
Secondary: 03C55: Set-theoretic model theory

Keywords
Peano arithmetic definability absoluteness

Citation

Shelah, Saharon; Tsuboi, Akito. Definability of Initial Segments. Notre Dame J. Formal Logic 43 (2002), no. 2, 65--73. doi:10.1305/ndjfl/1071509428. https://projecteuclid.org/euclid.ndjfl/1071509428


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References

  • Kaufmann, M., and J. H. Schmerl, "Remarks on weak notions of saturation in models of P"eano arithmetic, The Journal of Symbolic Logic, vol. 52 (1987), pp. 129–48.
  • Kirby, L. A. S., and J. B. Paris, "Initial segments of models of P"eano's axioms, pp. 211–26 in Set Theory and Hierarchy Theory V (Proceedings of the Third Conference on Set Theory and Hierarchy Theory, Bierutowice, 1976), Lecture Notes in Mathematics, vol. 619, edited by erseeditorsnames A. Lachlan and M. Srebrny and A. Zarach, Springer, Berlin, 1977.
  • Schmerl, J. H., "Some highly saturated models of P"eano arithmetic, The Journal of Symbolic Logic, vol. 67 (2002), pp. 1265–73.
  • Shelah, S., "Models with second-order properties. II". Trees with no undefined branches, Annals of Mathematical Logic, vol. 14 (1978), pp. 73–87.
  • Shelah, S., "Models with second order properties. IV". A general method and eliminating diamonds, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 183–212.
  • Shelah, S., Nonstructure Theory, to appear from Oxford University Press, Oxford, 2003.