Constant regions in models of arithmetic. (English) Zbl 1372.03085

Wong, Tin Lok

Notre Dame J. Formal Logic 56, No. 4, 603-624 (2015).

Summary: This paper introduces a new theory of constant regions, which generalizes that of interstices, in nonstandard models of arithmetic. In particular, we show that two homogeneity notions introduced by R. Kaye and the author [Ann. Pure Appl. Logic 161, No. 8, 987–1005 (2010; Zbl 1230.03070)], namely, constantness and pregenericity, are equivalent. This led to some new characterizations of generic cuts in terms of existential closedness.

MSC:

03C62	Models of arithmetic and set theory
03H15	Nonstandard models of arithmetic

Keywords:

constant regions; nonstandard models of arithmetic; interstices; generic cuts; existentially closed models

Citations:

Zbl 1230.03070

References:

[1]	Bamber, N., and H. Kotlarski, “On interstices of countable arithmetically saturated models of Peano arithmetic,” Mathematical Logic Quarterly , vol. 43 (1997), pp. 525-40. · Zbl 0884.03042
[2]	Bigorajska, T., H. Kotlarski, and J. H. Schmerl, “On regular interstices and selective types in countable arithmetically saturated models of Peano Arithmetic,” Fundamenta Mathematicae , vol. 158 (1998), pp. 125-46. · Zbl 0920.03069
[3]	Kaye, R., Models of Peano Arithmetic , vol. 15 of Oxford Logic Guides , Oxford University Press, New York, 1991. · Zbl 0744.03037
[4]	Kaye, R., “Generic cuts in models of arithmetic,” Mathematical Logic Quarterly , vol. 54 (2008), pp. 129-44. · Zbl 1138.03031
[5]	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 1118.03055
[6]	Kaye, R., and T. L. Wong, “Truth in generic cuts,” Annals of Pure and Applied Logic , vol. 161 (2010), pp. 987-1005. · Zbl 1230.03070
[7]	Kaye, R., and T. L. Wong, “The model theory of generic cuts,” to appear in Logic Without Borders: Essays on Set Theory, Model Theory, Philosophical Logic, and Philosophy of Mathematics , edited by J. Kennedy, R. Kossak, T. Hyttinen, A. Villaveces, and M. Kesälä, De Gruyter, Berlin.
[8]	Kirby, L. A. S., “Initial segments of models of arithmetic,” Ph.D. dissertation, University of Manchester, Manchester, UK, July 1977. · Zbl 0364.02032
[9]	Kossak, R., and J. H. Schmerl, The Structure of Models of Peano Arithmetic , vol. 50 of Oxford Logic Guides , Oxford University Press, Oxford, 2006. · Zbl 1101.03029
[10]	Schmerl, J. H., “Moving intersticial gaps,” Mathematical Logic Quarterly , vol. 48 (2002), pp. 283-96. · Zbl 0997.03036
[11]	Wong, T. L., “Generic cuts in a general setting,” Master’s thesis (qualifying), University of Birmingham, Birmingham, UK, 2008.