Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-13T07:02:56.032Z Has data issue: false hasContentIssue false
  • English
  • Français

Models with the ω-property

Published online by Cambridge University Press:  12 March 2014

Roman Kossak
Roman Kossak*
Affiliation:
Institute for Applications of Mathematics And Statistics, Agricultural University of Warsaw, 02-766 Warsaw, Poland Baruch College, City University of New York, New York, New York 10010
Get access Rights & Permissions [Opens in a new window]

Extract

In [KP] we have studied the problem of determining when a subset of a (countable) model M of PA can be coded in an elementary end extension of M. Sets with this property are called elementary extensional. In particular we can ask whether there are elementary extensional subsets of a model which have order type ω. It turns out that having elementary extensional subsets of order type ω is an interesting property connected with other structural properties of models of PA. We will call this property the ω-property. In [KP] the problem of characterizing models with the ω-property was left open. It is still open, and the aim of this paper is to present a collection of results pertaining to it. It should be mentioned that the same notion was studied by Kaufmann and Schmerl in [KS2] in connection with some weak notions of saturation which they discuss there. Our notion of a model with the ω-property corresponds to the notion of an upward monotonically ω-lofty cut.

It is fairly easy to see that countable recursively saturated models (or in fact all recursively saturated models with cofinality ω) and all short recursively saturated models have the ω-property (Proposition 1.2 below). On the other hand, if we had asked the question about the existence of models with the ω-property before 1975 (when recursively saturated models were introduced) the answer would probably not have been that easy and we would have to come to notions close to recursive saturation.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 1 , March 1989 , pp. 177 - 189
Copyright
Copyright © Association for Symbolic Logic 1989

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[H]Harrington, L., Building arithmetical models of PA, handwritten notes, University of California, Berkeley, California.Google Scholar
[JE]Jensen, D. and Ehrenfeucht, A., Some problems in elementary arithmetics, Fundamenta Mathematicae, vol. 92 (1976), pp. 223–245.CrossRefGoogle Scholar
[KS1]Kaufmann, M. and Schmerl, J., Saturation and simple extensions of models of Peano arithmetic, Annals of Pure and Applied Logic, vol. 27 (1984), pp. 109–136.CrossRefGoogle Scholar
[KS2]Kaufmann, M. and Schmerl, J., Remarks on weak notions of saturation in models of Peano arithmetic, this Journal, vol. 52 (1987), pp. 129–148.Google Scholar
[Kir]Kirby, L. A. S., Initial segments in models of arithmetic, Ph.D. thesis, University of Manchester, Manchester, 1977.Google Scholar
[Kos]Kossak, R., A certain class of models of arithmetic, this Journal, vol. 49 (1983), pp. 311–320.Google Scholar
[Kos1]Kossak, R., A note on satisfaction classes, Notre Dame Journal of Formal Logic, vol. 26 (1985), pp. 1–8.CrossRefGoogle Scholar
[Kos2]Kossak, R., Recursively saturated ω 1-like models of arithmetic, Notre Dame Journal of Formal Logic, vol. 26 (1985), pp. 413–422.CrossRefGoogle Scholar
[KK]Kossak, R. and Kotlarski, H., Results on automorphisms of recursively saturated models of PA, Fundamenta Mathematicae (to appear).Google Scholar
[KP]Kossak, R. and Paris, J., Subsets of models of arithmetic, Open days in model theory and set theory (Jadwisin, Poland, 1981; Guzicki, W.et al., editors), University of Leeds, Leeds, England, 1984, pp. 159–179.Google Scholar
[Kot]Kotlarski, H., On elementary cuts in recursively saturated models of Peano arithmetic, Fundamenta Mathematicae, vol. 120 (1984), pp. 206–222.CrossRefGoogle Scholar
[Ku]Kueker, D., Back and forth arguments and infinitary logics, Infinitary logic: in memoriam Carol Karp, Lecture Notes in Mathematics, vol. 492, Springer-Verlag, Berlin, 1975, pp. 17–73.CrossRefGoogle Scholar
[M]Marker, D., Degrees coding models of arithmetic, Ph.D. thesis, Yale University, New Haven, Connecticut, 1983.Google Scholar
[PK]Paris, J. and Kirby, L., Σn-collection schemas in arithmetic, Logic Colloquium '77 (Macintyre, A.et al., editors), North-Holland, Amsterdam, 1978, pp. 199–209.CrossRefGoogle Scholar
[Smo1]Smoryński, C., Elementary extensions of recursively saturated models of arithmetic, Notre Dame Journal of Formal Logic, vol. 23 (1982), pp. 193–203.CrossRefGoogle Scholar
[Smo2]Smoryński, C., A note on initial segment constructions in recursively saturated models of arithmetic, Notre Dame Journal of Formal Logic, vol. 23 (1982), pp. 393–408.CrossRefGoogle Scholar
[SS]Smoryński, C. and Stavi, J., Cofinal extension preserves recursive saturation, Model theory of algebra and arithmetic (proceedings, Karpacz, 1979; Pacholski, L.et al., editors), Lecture Notes in Mathematics, vol. 834, Springer-Verlag, Berlin, 1980, pp. 338–345.Google Scholar