Four problems concerning recursively saturated models of arithmetic. (English) Zbl 0848.03016

The paper presents four open problems. One concerns a possible converse to Tarski’s undefinability of truth theorem, and is of 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.


03C62 Models of arithmetic and set theory
03C50 Models with special properties (saturated, rigid, etc.)
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[1] Gaifman, H., “Models and types of Peano’s Arithmetic,” Annals of Mathematical Logic , vol. 9 (1976), pp. 223–306. · Zbl 0332.02058
[2] Kaye, R., Models of Peano Arithmetic , Oxford Logic Guides, Oxford University Press, Oxford, 1991. · Zbl 0744.03037
[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. · Zbl 0748.03023
[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. · Zbl 0364.02032
[6] Knight, J. F., “Hanf number for omitting types over particular theories,” The Journal of Symbolic Logic , vol. 41 (1976), pp. 583–588. JSTOR: · Zbl 0343.02039
[7] Kossak, R., “A certain class of models of Peano Arithmetic,” The Journal of Symbolic Logic , vol. 48 (1983), pp. 311–320. JSTOR: · Zbl 0514.03036
[8] Kossak, R., “Remarks on free sets,” Bulletin of the Polish Academy of Sciences , vol. 34 (1986), pp. 117–122. · Zbl 0623.03056
[9] Kossak, R., and H. Kotlarski, “Results on automorphisms of recursively saturated models of \PA,” Fundamenta Mathematicæ , vol. 129 (1988), pp. 9–15. · Zbl 0662.03027
[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. · Zbl 0796.03043
[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. · Zbl 0748.03024
[12] Kotlarski, H., “Full satisfaction classes: a survey,” Notre Dame Journal of Formal Logic , vol. 32 (1991), pp. 573–579. · Zbl 0752.03018
[13] Smoryński, C., “Elementary extensions of \rs models of arithmetic,” Notre Dame Journal of Formal Logic , vol. 22 (1981), pp. 193–203. · Zbl 0503.03032
[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. · Zbl 0519.03055
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