## 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 -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

http://projecteuclid.org/euclid.ndjfl/1040136913

Digital Object Identifier
doi:10.1305/ndjfl/1040136913

Mathematical Reviews number (MathSciNet)
MR1368464

Zentralblatt MATH identifier
0848.03016

#### 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.

#### 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.