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Model Theory: Geometrical and Set-Theoretic Aspects and Prospects

Published online by Cambridge University Press:  15 January 2014

Angus Macintyre
Angus Macintyre*
Affiliation:
Department of Mathematics and Statistics, University of Edinburgh, Edinburgh, EH93JZ, UK.E-mail:angus@maths.ed.ac.uk
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Extract

I see model theory as becoming increasingly detached from set theory, and the Tarskian notion of set-theoretic model being no longer central to model theory. In much of modern mathematics, the set-theoretic component is of minor interest, and basic notions are geometric or category-theoretic. In algebraic geometry, schemes or algebraic spaces are the basic notions, with the older “sets of points in affine or projective space” no more than restrictive special cases. The basic notions may be given sheaf-theoretically, or functorially. To understand in depth the historically important affine cases, one does best to work with more general schemes. The resulting relativization and “transfer of structure” is incomparably more flexible and powerful than anything yet known in “set-theoretic model theory”.

It seems to me now uncontroversial to see the fine structure of definitions as becoming the central concern of model theory, to the extent that one can easily imagine the subject being called “Definability Theory” in the near future.

Tarski's set-theoretic foundational formulations are still favoured by the majority of model-theorists, and evolution towards a more suggestive language has been perplexingly slow. None of the main texts uses in any nontrivial way the language of category theory, far less sheaf theory or topos theory. Given that the most notable interactions of model theory with geometry are in areas of geometry where the language of sheaves is almost indispensable (to the geometers), this is a curious situation, and I find it hard to imagine that it will not change soon, and rapidly.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 9 , Issue 2 , June 2003 , pp. 197 - 212
Copyright
Copyright © Association for Symbolic Logic 2003

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References

REFERENCES

[1] Ax, J., The elementary theory of finite fields, Annals of Mathematics. Second Series, vol. 88 (1968), pp. 239–271.CrossRefGoogle Scholar
[2] Buium, A., Arithmetic analogues of derivations, Journal of Algebra, vol. 198 (1997), no. 1, pp. 290–299.CrossRefGoogle Scholar
[3] Chatzidakis, Z. and Hrushovski, E., Model theory of difference fields, Transactions of the American Mathematical Society, vol. 351 (1999), no. 8, pp. 2997–3071.CrossRefGoogle Scholar
[4] Cherlin, G., Harrington, L., and Lachlan, A.H., ℵ0-categorical, ℵ0-stable structures, Annals of Pure and Applied Logic, vol. 28 (1985), no. 2, pp. 103–135.Google Scholar
[5] Deligne, P., Quelques idées maîtresses de l'œuvre de A. Grothendieck, Matériaux pour l'histoire des mathématiques au XXième siècle (Nice 1996), 11–19, Sémin. Congr. 3 (Paris), Soc. Math. France, 1998.Google Scholar
[6] Denef, J. and Loeser, F., Definable sets, motives and p-adic integrals, Journal of the American Mathematical Society, vol. 14 (2001), no. 2, pp. 429–469.Google Scholar
[7] Denef, J. and van den Dries, L., p-adic and real subanalytic sets, Annals of Mathematics. Second Series, vol. 128 (1988), no. 1, pp. 79–138.Google Scholar
[8] Fried, M., Haran, D., and Jarden, M., Galois stratification over Frobenius fields, Advances in Mathematics, vol. 51 (1984), no. 1, pp. 1–35.Google Scholar
[9] Grothendieck, A., Esquisse d'un programme, London Mathematical Society Lecture Notes Series 242, Geometric Galois actions, vol. 1, Cambridge University Press, 1997, pp. 5–48.Google Scholar
[10] Hrushovski, E., The Mordell-Lang conjecture for function fields, Journal of the American Mathematical Society, vol. 9 (1996), no. 3, pp. 667–690.Google Scholar
[11] Hrushovski, E., Geometric model theory, Documenta Math, Proceedings ICM 98 (Berlin), vol. 1, June 1998, pp. 281–302.Google Scholar
[12] Hrushovski, E., Lecture at MSRI, 06 1998.Google Scholar
[13] Hrushovski, E., The Manin-Mumford conjecture and the model theory of difference fields, Annals of Pure and Applied Logic, vol. 112 (2001), no. 1, pp. 43–115.Google Scholar
[14] Hrushovski, E., The first order theory of the Frobenius automorphism, preprint.Google Scholar
[15] Hrushovski, E. and Zilber, B., Zariski geometries, Journal of the American Mathematical Society, vol. 9 (1996), no. 1, pp. 1–56.Google Scholar
[16] Karpinski, M. and Macintyre, A., Polynomial bounds for VC-dimension of sigmoidal and general Pfaffian neural networks, Journal ofComputer and System Sciences, vol. 54 (1997), no. 1 part 2, pp. 169–176.Google Scholar
[17] Karpinski, M. and Macintyre, A., Approximating volumes and integrals in o-minimal and p-minimal structures, Quaderni di Matematica, vol. 6 (2001), pp. 151–177, Connections between Model Theory and Algebraic and Analytic Geometry.Google Scholar
[18] Kim, B. and Pillay, A., From stability to simplicity, this Bulletin, vol. 4 (1998), no. 1, pp. 17–36.Google Scholar
[19] Kreisel, G. and Macintyre, A., Constructive logic versus algebraization 1, The L.E.J. Brouwer Centenary Symposium, North Holland, 1982, pp. 217–260.Google Scholar
[20] Laskowski, M.C., Vapnik-Chervonenkis classes of definable sets, Journal of the London Mathematical Society. Second Series, vol. 45 (1992), no. 2, pp. 377–384.Google Scholar
[21] Lawvere, F. W., Quantifiers and sheaves, Actes Congrès. Intern. Math. 1970, Gauthiers-Villars, 1971, pp. 329–334.Google Scholar
[22] Macintyre, A., Generic automorphisms of fields, Annals of Pure and Applied Logic, vol. 88 (1997), no. 2–3, pp. 165–180.Google Scholar
[23] Macintyre, A., Axioms for Frobenius, in preparation.Google Scholar
[24] Macintyre, A. and Wilkie, A. J., On the decidability of the real exponential field, Kreiseliana (Odifreddi, P., editor), A K Peters, Wellesley, MA, 1996, pp. 441–467.Google Scholar
[25] Manin, Y. I., Lectures on zeta functions and motives (according to Deninger and Kurokawa), Astérisque, vol. 228 (1995), pp. 121–163.Google Scholar
[26] Matsumura, H., Commutative algebra, Mathematics Lecture Note Series vol. 56, Benjamin, 2nd ed., 1980.Google Scholar
[27] Morley, M. D., Categoricity in power, Transactions of the American Mathematical Society, vol. 114 (1965), pp. 514–538.Google Scholar
[28] Morley, M. D., Applications of topology to ω1, ω , Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math. vol. XXV, University of California, 1971), American Mathematical Society, 1974, pp. 233–240.Google Scholar
[29] Pour La Science, Les Génies de la Science, vol. 2, p. 70, Bourbaki, 2000, p. 70.Google Scholar
[30] Sela, Z., numerous preprints 2001.Google Scholar
[31] van den Dries, L., Tame topology and o-minimal structure, London Mathematical Society Lecture Note Series 248, Cambridge University Press, 1998.Google Scholar
[32] van den Dries, L. and Macintyre, A., The logic of Rumely's local-global principle, Journal für die Reine und Angewandte Mathematik, vol. 407 (1990), pp. 33–56.Google Scholar
[33] van den Dries, L., Macintyre, A., and Marker, D., The elementary theory of restricted analytic fields with exponentiation, Annals of Mathematics. Second Series, vol. 140 (1994), no. 1, pp. 183–205.Google Scholar
[34] van den Dries, L., Logarithmic-exponential series, Proceedings of the international conference “Analyse & Logique” (Mons, 1997), Annals of Pure and Applied Logic, vol. 111 (2001), no. 1–2, pp. 61–113.Google Scholar
[35] van den Dries, L. and Miller, C., Geometric categories and o-minimal structures, Duke Mathematical Journal, vol. 84 (1996), no. 2, pp. 497–540.Google Scholar
[36] Wilkie, A. J., Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, Journal of the American Mathematical Society, vol. 9 (1996), no. 4, pp. 1051–1094.Google Scholar
[37] Wilkie, A. J., A theorem of the complement and some new o-minimal structures, Selecta Mathematica, New Series, vol. 5 (1999), no. 4, pp. 397–421.Google Scholar
[38] Zilber, B., Analytic and pseudoanalytic structures, preprint, Oxford, 2001.Google Scholar