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The theory of \(\kappa\)-like models of arithmetic. (English) Zbl 0848.03019
A linearly ordered model is \(\kappa\)-like if it is of power \(\kappa\), and each of its proper initial segments is of cardinality smaller than \(\kappa\). In the search for the theory of all \(\kappa\)-like models of PA, the author describes axiom schemes true in all \(\kappa\)-like models of PA, and discusses their model-theoretic properties. The schemes are suitable formalizations of: the statement that in a \(\kappa\)-like model there is no one-to-one map from the model to a proper initial segment; various forms of the pigeonhole principle; collection scheme; an indiscernibility schema related to a general theorem of Keisler on \(\kappa\)-like models for singular strong limits \(\kappa\). The concept of tree indiscernibility related to the work of Paris and Mills on cardinalities of initial segments of models of PA is also discussed. The paper is intended to be read in conjunction with another paper of the author’s [“Constructing \(\kappa\)-like models of arithmetic”, J. Lond. Math. Soc. (to appear)].

03C62 Models of arithmetic and set theory
03F30 First-order arithmetic and fragments
Full Text: DOI
[1] Gaifman, H., “A note on models and submodels of arithmetic,” pp. 128–44 in Proceedings of the Conference on Mathematical Logic, London 1970 , Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1972. · Zbl 0255.02058
[2] Kaye, R., “Model-theoretic properties characterizing Peano arithmetic,” The Journal of Symbolic Logic , vol. 56 (1991), pp. 949–963. JSTOR: · Zbl 0746.03032
[3] Kaye, R., Models of Peano Arithmetic , Vol. 15 of Oxford Logic Guides , Oxford University Press, Oxford, 1991. · Zbl 0744.03037
[4] Kaye, R., “On cofinal extensions of models of fragments of arithmetic,” Notre Dame Journal of Formal Logic , vol. 32 (1991), pp. 399–408. · Zbl 0746.03033
[5] Kaye, R., “Constructing \(\kappa\)-like models of arithmetic,” forthcoming in Journal of the London Mathematical Society . · Zbl 0865.03056
[6] Keisler, H. J., “Models with orderings,” pp. 35–62 in Logic, Methodology and Philosophy of Science III , edited by B. van Rootselaar and J. F. Staal, North-Holland, Amsterdam, 1968. · Zbl 0191.29503
[7] 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, Bierutowice, Poland, 1976 , Vol. 619 of Lecture Notes in Mathematics , edited by A. H. Lachlan et al., Springer-Verlag, Berlin, 1977. · Zbl 0364.02032
[8] Lessan, H., Models of Arithmetic , Ph.D. thesis, University of Manchester, 1978.
[9] Paris, J. B., and L. A. S. Kirby, “\(\Sigma_n\) collection schemas in arithmetic,” pp. 199–209 in Logic Colloquium ’77 , edited by A. J. Macintyre et al., North-Holland, Amsterdam, 1978. · Zbl 0442.03042
[10] Paris, J. B., and G. Mills, “Closure properties of countable non-standard integers,” Fundamenta Mathematicæ , vol. 103 (1979), pp. 205–215. · Zbl 0421.03051
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