Notre Dame Journal of Formal Logic

Four Problems Concerning Recursively Saturated Models of Arithmetic

Roman Kossak

Abstract

The paper presents four open problems. One concerns a possible converse to Tarski's undefinability of truth theorem, and is of a general character. The other three are more specific. The questions are about some special $\omega_1$-like models, initial segments of countable recursively saturated models of PA, and about extendability of automorphisms. In each case a partial answer is given. All partial solutions are based on applications of inductive satisfaction classes.

Article information

Source
Notre Dame J. Formal Logic Volume 36, Number 4 (1995), 519-530.

Dates
First available in Project Euclid: 17 December 2002

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1040136913

Mathematical Reviews number (MathSciNet)
MR1368464

Digital Object Identifier
doi:10.1305/ndjfl/1040136913

Zentralblatt MATH identifier
0848.03016

Subjects
Primary: 03C62: Models of arithmetic and set theory [See also 03Hxx]
Secondary: 03C57: Effective and recursion-theoretic model theory [See also 03D45]

Citation

Kossak, Roman. Four Problems Concerning Recursively Saturated Models of Arithmetic. Notre Dame J. Formal Logic 36 (1995), no. 4, 519--530. doi:10.1305/ndjfl/1040136913. http://projecteuclid.org/euclid.ndjfl/1040136913.


Export citation

References

  • [1] Gaifman, H., ``Models and types of Peano's Arithmetic,'' Annals of Mathematical Logic, vol. 9 (1976), pp. 223--306.
  • [2] Kaye, R., Models of Peano Arithmetic, Oxford Logic Guides, Oxford University Press, Oxford, 1991.
  • [3] 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.
  • [4] Kirby, L. A. S., Ph.D. thesis, University of Manchester, 1977.
  • [5] Kirby, L. A. S., and J. B. Paris, ``Initial segments of models of Peano's axioms,'' pp. 211--226 in Set Theory and Hierarchy Theory V, Lecture Notes in Mathematics 619, Springer-Verlag, Berlin, 1977.
  • [6] Knight, J. F., ``Hanf number for omitting types over particular theories,'' The Journal of Symbolic Logic, vol. 41 (1976), pp. 583--588.
  • [7] Kossak, R., ``A certain class of models of Peano Arithmetic,'' The Journal of Symbolic Logic, vol. 48 (1983), pp. 311--320.
  • [8] Kossak, R., ``Remarks on free sets,'' Bulletin of the Polish Academy of Sciences, vol. 34 (1986), pp. 117--122.
  • [9] Kossak, R., and H. Kotlarski, ``Results on automorphisms of recursively saturated models of \PA,'' Fundamenta Mathematicæ, vol. 129 (1988), pp. 9--15.
  • [10] Kossak, R., H. Kotlarski and J. H. Schmerl, ``On maximal subgroups of the automorphism group of a countable \rs model of \PA,'' Annals of Pure and Applied Logic, vol. 65 (1993), pp. 125--148.
  • [11] Kossak, R., and J. H. Schmerl, ``Minimal satisfaction classes with an application to rigid models of Peano Aritmetic,'' Notre Dame Journal of Formal Logic, vol. 32 (1991), pp. 392--398.
  • [12] Kotlarski, H., ``Full satisfaction classes: a survey,'' Notre Dame Journal of Formal Logic, vol. 32 (1991), pp. 573--579.
  • [13] Smoryński, C., ``Elementary extensions of \rs models of arithmetic,'' Notre Dame Journal of Formal Logic, vol. 22 (1981), pp. 193--203.
  • [14] Smoryński, C., ``A note on initial segment constructions in \rs models of arithmetic,'' Notre Dame Journal of Formal Logic, vol. 23 (1982), pp. 393--408.