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A note on generic projective planes. (English) Zbl 1050.03025

Summary: E. Hrushovski (1988) constructed an \(\omega\)-categorical stable pseudoplane which refuted Lachlan’s conjecture. In this note, we show that an \(\omega\)-categorical projective plane cannot be constructed by “the Hrushovski method.”

MSC:

03C45 Classification theory, stability, and related concepts in model theory
03C35 Categoricity and completeness of theories
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References:

[1] Baldwin, J. T., ”Problems on ‘pathological’ structures”, pp. 1–9 in Proceedings of the Tenth Easter Conference on Model Theory , edited by M. Weese and H. Wolter, vol. 93 of Seminarberichte , Humboldt Universität Fachbereich Mathematik, Berlin, 1993. · Zbl 0798.03035
[2] Baldwin, J. T., ”An almost strongly minimal non-Desarguesian projective plane”, Transactions of the American Mathematical Society , vol. 342 (1994), pp. 695–711. · Zbl 0796.03041
[3] Baldwin, J. T., and N. Shi, ”Stable generic structures”, Annals of Pure and Applied Logic , vol. 79 (1996), pp. 1–35. · Zbl 0857.03020
[4] Cameron, P. J., Oligomorphic Permutation Groups , vol. 152 of the London Mathematical Society Lecture Note Series , Cambridge University Press, Cambridge, 1990. · Zbl 0813.20002
[5] Cherlin, G., L. Harrington, and A. H. Lachlan, ”\(\aleph_ 0\)”-categorical, \(\aleph_ 0\)-stable structures, Annals of Pure and Applied Logic , vol. 28 (1985), pp. 103–35. · Zbl 0566.03022
[6] Hodges, W., Model Theory , vol. 42 of Encyclopedia of Mathematics and its Applications , Cambridge University Press, Cambridge, 1993. · Zbl 0789.03031
[7] Hrushovski, E., ”A stable \(\aleph_0\)-categorical pseudoplane”, preprint, 1988.
[8] Wagner, F. O., ”Relational structures and dimensions”, pp. 153–80 in Automorphisms of First-Order Structures , Oxford University Press, New York, 1994. · Zbl 0813.03020
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