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COMPLETENESS AND CATEGORICITY (IN POWER): FORMALIZATION WITHOUT FOUNDATIONALISM*

Published online by Cambridge University Press:  13 May 2014

JOHN T. BALDWIN
JOHN T. BALDWIN*
Affiliation:
DEPARTMENT OF MATHEMATICS, STATISTICS AND COMPUTER SCIENCE M/C 249, UNIVERSITY OF ILLINOIS AT CHICAGO, CHICAGO, IL 6067, USA
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Abstract

We propose a criterion to regard a property of a theory (in first or second order logic) as virtuous: the property must have significant mathematical consequences for the theory (or its models). We then rehearse results of Ajtai, Marek, Magidor, H. Friedman and Solovay to argue that for second order logic, ‘categoricity’ has little virtue. For first order logic, categoricity is trivial; but ‘categoricity in power’ has enormous structural consequences for any of the theories satisfying it. The stability hierarchy extends this virtue to other complete theories. The interaction of model theory and traditional mathematics is examined by considering the views of such as Bourbaki, Hrushovski, Kazhdan, and Shelah to flesh out the argument that the main impact of formal methods on mathematics is using formal definability to obtain results in ‘mainstream’ mathematics. Moreover, these methods (e.g., the stability hierarchy) provide an organization for much mathematics which gives specific content to dreams of Bourbaki about the architecture of mathematics.

Keywords

Categoricityformalizationmodel theoryfoundationalismcompleteness
Type
Articles
Information
Bulletin of Symbolic Logic , Volume 20 , Issue 1 , March 2014 , pp. 39 - 79
Copyright
Copyright © The Association for Symbolic Logic 2014 

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Footnotes

*

I realized while writing that the title was a subconscious homage to the splendid historical work on Completeness and Categoricity by Awodey and Reck [6].

References

REFERENCES

Ajtai, M., Isomorphism and higher order equivalence. Annals of Mathematical Logic, vol. 16. 1979:181–203.Google Scholar
Aldama, Ricardo de, Definable nilpotent and soluble envelopes in groups without the independence property. Mathematical Logic Quarterly, vol. 59 (2013).Google Scholar
Altinel, T. and Baginski, P., Definable envelopes of nilpotent subgroups of groups with chain conditions on centralizers. Proceedings of the American Mathematical Society, vol. 142 (2014),pp. 1497–1506.Google Scholar
Altinel, T., Borovik, A., and Cherlin, G., Simple groups of finite Morley rank, American Mathematical Society Monographs Series. American Mathematical Society, Providence, RI, 2008.Google Scholar
Avigad, J., Review of the birth of model theory: Löwenheim’s theorem in the frame of the theory of relatives. The Mathematical Intelligencer, (2006), pp. 67–71.CrossRefGoogle Scholar
Awodey, S. and Reck, E., Completeness and categoricity, part I: Nineteenth-century axiomatics to twentieth-century metalogic. History and Philosophy of Logic, vol. 23 (2002), pp. 1–30.CrossRefGoogle Scholar
Awodey, S. and Reck, E., Completeness and categoricity, part II: Twentieth-century metalogic to twenty-first century semantics. History and Philosophy of Logic, vol. 23 (2002), pp. 77–94.Google Scholar
Ax, James, The elementary theory of finite fields. Annals of Mathematics, vol. 88 (1968), pp. 239–271.Google Scholar
Badesa, C., The birth of model theory, Princeton University Press, Princeton, NJ, 2004.Google Scholar
Baginski, P., Stable א 0-categorical alternative rings. 2012, preprint.Google Scholar
Baldwin, T., Categoricity, Number 51 in University Lecture Notes. American Mathematical Society, Providence, RI, 2009, www.math.uic.edu/~jbaldwin.Google Scholar
Baldwin, T., A field guide to Hrushovski constructions. Report: http://www.math.uic.edu/∼jbaldwin/pub/hrutrav.pdf.Google Scholar
Baldwin, T., Stability theory and algebra. Journal of Symbolic Logic, vol. 44 (1979), pp. 599–608.Google Scholar
Baldwin, T., Notes on quasiminimality and excellence., the Journal, vol. 10 (2004), pp. 334–367.Google Scholar
Baldwin, T., Formalization, primitive concepts, and purity. Review of Symbolic Logic,(2013),pp. 87–128. http://homepages.math.uic.edu/∼jbaldwin/pub/purityandvocab10.pdf.Google Scholar
Baldwin, J.T. and Lachlan, A.H., On strongly minimal sets. Journal of Symbolic Logic, vol. 36 (1971), pp. 79–96.Google Scholar
Baldwin, J.T. and Rose, B., א0 -categoricity and stability of ring. Journal of Algebra, vol. 45 (1977), pp. 1–17.Google Scholar
Baldwin, J.T. and Shelah, S., Stability spectrum for classes of atomic models. Journal of Mathematical Logic, vol. 12 (2012), preprint, http://www.math.uic.edu/\∼jbaldwin/pub/shnew22.Google Scholar
Barwise, P. and Eklof, J., Lefschetz’s principle. Journal of Algebra, vol. 13 (1969), pp. 554–570.Google Scholar
Beeson, M., Constructive geometry, proof theory and straight-edge and compass constructions. http://www.michaelbeeson.com/research/talks/ConstructiveGeometrySlides.pdf.Google Scholar
Białynicki-Birula, A. and Rosenlicht, M., Injective morphisms of real algebraic varieties. Proceedings of the American Mathematical Society, vol. 13 (1962), pp. 200–203.Google Scholar
Blum, L., Generalized algebraic structures: A model theoretical approach., PhD thesis, MIT, 1968.Google Scholar
Borovik, A. and Nesin, A., Groups of finite Morley rank. Oxford University Press, Oxford, UK, 1994.Google Scholar
Bourbaki, Nicholas, The architecture of mathematics. The American Mathematical Monthly, vol. 57 (1950), pp. 221–232.CrossRefGoogle Scholar
Bouscaren, E., editor. Model theory and algebraic geometry : An introduction to E. Hrushovski’s proof of the geometric Mordell-Lang conjecture, Springer-Verlag, Heidelberg, 1999.Google Scholar
Cherlin, G.L., Algebraicity conjecture. webpage Algebraicityconjecture, 2004.Google Scholar
Cherlin, G.L., Harrington, L., and Lachlan, A.H., א0-categorical, א0-stable structures. Annals of Pure and Applied Logic, vol. 28 (1985), pp. 103–135.CrossRefGoogle Scholar
Cherlin, G.L. and Hrushovski, E., Finite structures with few types, Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, 2003.Google Scholar
Corcoran, John, Categoricity. History and philosophy of logic, vol. 1 (1980), pp. 187–207. http://dx.doi.org/10.1080/01445348008837010.CrossRefGoogle Scholar
Van den Dries, L., Tame topology and O-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, 1999.Google Scholar
Detlefsen, M., Completeness and the ends of axiomatization., Interpreting Goedel (Kennedy, J., editor), Cambridge University Press, 2014.Google Scholar
Dieudonné, Jean. Lés methodes axiomatique modernes et les fondements des mathématique. Revue Scientique, vol. 77 (1939), pp. 224–232.Google Scholar
Ehrlich, P., The absolute arithmetic continnum and the unification of all numbers great and small, this Journal, vol. 18 (2012), pp. 1–45.Google Scholar
Eklof, P., Lefschetz’s principle and local functors. Proceedings of the American Mathematical Society, vol. 37 (1973), pp. 333–339.Google Scholar
Evans, D., Homogeneous geometries. Proceedings of the London Mathematical Society, vol. 52 (1986), pp. 305–327.CrossRefGoogle Scholar
Fraenkel, A., Einleitung in die Mengenlehre., Springer, Berlin, 1928. 3rd, revised edition.Google Scholar
Fraïssé, R.Deux relations dénombrables, logiquement équivalentes pour le second ordre, sont isomorphes (modulo un axiome de constructibilité., Mathematical logic and formal systems, Lecture Notes in Pure and Applied Mathematics, vol. 94, Dekker, New York, 1985, pp. 161–182.Google Scholar
Freitag, J., Isogeny in supertable groups. 2012, preprint, http://arxiv.org/abs/1106.0695.Google Scholar
Freitag, J., Isogeny in supertable groups. Archive for Mathematical Logic, to appear, preprint,http://arxiv.org/abs/1110.1766.Google Scholar
Grossberg, R. and Hart, Bradd, The classification theory of excellent classes. The Journal of Symbolic Logic, vol. 54 (1989), pp. 1359–1381.Google Scholar
Grothendieck, A., Elements de géometrie algébriques (rédigés avec la collaboration de J. Dieudonné), IV Etudes Locale des Schémas et des Morphismes de Schémas. Publications mathématiques de l’IHÉS, vol. 28 (1966), pp. 5–255.Google Scholar
Hafner, J. and Mancosu, P., Beyond unification, The Philosophy of Mathematical Practice (Mancosu, P., editor), Oxford University Press, 2008, pp. 151–178.Google Scholar
Hart, B., Hrushovski, E, and Laskowski, C., The uncountable spectra of countable theories. Annals of Mathematics, vol. 152 (2000), pp. 207–257.Google Scholar
Haskell, D., Hrushovski, E., and MacPherson, H.D., Stable domination and independence in algebraically closed valued fields., Lecture Notes in Logic. Association for Symbolic Logic, 2007.Google Scholar
Howard, W., Comments on the relations of Bourbaki and logicians. 2013, http://homepages.math.uic.edu/∼jbaldwin/pub/howonbour.pdf.Google Scholar
Hrushovski, E., A new strongly minimal set. Annals of Pure and Applied Logic, vol. 62 (1993), pp. 147–166.CrossRefGoogle Scholar
Hrushovski, E., Stability and its uses, Current developments in mathematics, 1996 (Cambridge, MA), International Press, Boston, MA, 1997, pp. 61–103.Google Scholar
Hrushovski, E., Geometric model theory, Proceedings of the International Congress of Mathematicians Vol. I (Berlin, 1998), 1998, pp. 281–302.Google Scholar
Hrushovski, E. and Zilber, B., Zariski geometries. Bulletin of the American Mathematical Society, vol. 28 (1993), pp. 315–324.Google Scholar
Hrushovski, Ehud, Almost orthogonal regular types. Annals of Pure and Applied Logic, vol. 45 (1989), pp. 139–155.Google Scholar
Hyttinen, Tapani, Kangas, Kaisa, and Väänänen, Jouko, On second order characterizability. Logic Journal of the IGPL, vol. 21 (2013), 767–787, arXiv:1208.5167, doi 10.1093/jigpal/jzs047.Google Scholar
Jacobson, N., Lectures in Abstract Algebra III, University Series in Higher Mathematics. Van Nostrand, Princeton, NJ, 1964.CrossRefGoogle Scholar
Kang, M.C., Injective morphisms of affine varieties.Proceedings of the American Mathematical Society, vol. 119 (1993), pp. 1–4.Google Scholar
Kazhdan, David, Lecture notes in motivic integration: Logic., 2006, http://www.ma.huji.ac.il/∼kazhdan/Notes/motivic/b.pdf.Google Scholar
Kennedy, Juliette, On formalism freeness., this Journal, vol. 19 (2013), pp. 351–393.Google Scholar
Keskinen, L., Characterizing all models in infinite cardinalities. Annals of Pure and Applied Logic, vol. 164 (2008), pp. 230–250.Google Scholar
Kojman, M. and Shelah, S., Non-existence of universal orders in many cardinals. Journal of Symbolic Logic, vol. 875–891 (1992), pp. 261–294. paper KhSh409.Google Scholar
Kolchin, E., Differential algebra & algebraic groups, Academic Press, New York, 1973.Google Scholar
Lindstrom, P., On model completeness. Theoria, vol. 30 (1964), pp. 183–196.Google Scholar
Macintyre, Angus J., The impact of Gödel’s incompleteness theorems on mathematics, Kurt Gödel and the foundations of mathematics (Baaz, et al. ., editors), Cambridge University Press, Cambridge, UK, 2011, pp. 3–26.Google Scholar
Macpherson, D. and Steinhorn, C., Definability in classes of finite structures., Finite and algorithmic model theory (Esparza, Javier, Michaux, C., and Steinhorn, C., editors), London Mathematical Society Lecture Note Series, vol. 379, Cambridge University Press, Cambridge, UK, 2011, pp. 140–176.Google Scholar
Malcev, A.I., A general method for obtaining local theorems in group theory, The metamathematics of algebraic systems, collected papers: 1936–1967 Wells, B.F., editor), North-Holland Publication Co., Amsterdam, 1971, pp. 1–14 1941 Russian original in Uceny Zapiski Ivanov Ped. Inst.Google Scholar
Marek, Wiktor, Consistance d’une hypothèse de Fraïssè sur la définissabilité dans un langage du second ordre. Comptes Rendus de l’Académie des Sciences. Série A-B. vol. 276 (1973), pp. A1147–A1150.Google Scholar
Marker, D., Review of: Tame topology and o-minimal structures, by Lou van den Dries. Bulletin of the American Mathematical Society, vol. 37 (2000), pp. 351–357.Google Scholar
Marker, D., Model theory: An introduction, Springer-Verlag, New York, 2002.Google Scholar
Marker, D., The number of countable differentially closed fields. Notre Journal of Formal Logic, vol. 48 (2007), pp. 99–113.Google Scholar
Marker, D., Messmer, M., and Pillay, A., Model theory of fields, Springer-Verlag, Berlin, 1996.Google Scholar
Mathias, A.R.D., The ignorance of Bourbaki. Mathematical Intelligencer, vol. 14 (1992), pp. 4–13. also in: Physis Riv. Internaz. Storia Sci (N.S.) vol. 28(1991), pp. 887–904.Google Scholar
Mathias, A.R.D., Hilbert, Bourbaki, and the scorning of logic. 2012, preprint. https://www.dpmms.cam.ac.uk/∼ardm/logbanfinalmk.pdf.Google Scholar
Morley, M., Categoricity in power. Transactions of the American Mathematical Society, vol. 114 (1965), pp. 514–538.CrossRefGoogle Scholar
Morley, M., Omitting classes of elements., The theory of models (Addison, Henkin, and Tarski, , editors), North-Holland, Amsterdam, 1965, pp. 265–273.Google Scholar
Peretyatkin, M. G., Finitely axiomatizable theories., Springer-Verlag, New York, 1997.Google Scholar
Pillay, A., On groups and fields definable in o-minimal structures. Journal of Pure and Applied Algebra, vol. 55 (1988), pp. 239–255.Google Scholar
Pillay, A., Model theory. Logic and Philosophy today, Part 1: Journal of Indian Council of Philosophical Research, 28, 2010. http://www1.maths.leeds.ac.uk/∼pillay/modeltheory.jicpr.revised.pdf; reprinted in Logic and Philosophy Today, Studies in Logic 29 (& 30), College Publications, London, 2011.Google Scholar
Poizat, Bruno, Groupes Stables. Nur Al-mantiq Wal-ma’rifah, 82, Rue Racine 69100 Villeurbanne France, 1987.Google Scholar
Poizat, Bruno, Stable groups., American Mathematical Society, Providence, RI, 2001. translated from Groupes Stables.CrossRefGoogle Scholar
Post, E., Introduction to a general theory of elementary propositions, From Frege to Godel: A sourcebook in mathematical logic, 1879–1931 (Van Heijenoort, J., editor), Harvard University Press, Cambridge, MA, 1967, pp. 264–283.Google Scholar
Robinson, A., On predicates in algebraically closed fields. The Journal of Symbolic Logic, vol. 19 (1954), pp. 103–114.Google Scholar
Ryll-Nardzewski, C., On categoricity in power ≤ א0. Bulletin L’Académie Polonaise des Science, Série des Sciences Mathématiques, Astronomiques et Physiques, vol. 7 (1959), pp. 545–548.Google Scholar
Sacks, Gcerald E., Remarks against foundational activity. Historia Mathematica, vol. 2 (1975), pp. 523–528.Google Scholar
Scanlon, , Motivic integration: An outsider’s tutorial, Lecture Durham, England, 2009. http://math.berkeley.edu/∼scanlon/papers/scanlon_durham_motivic_integration_outsiders_tutorial.pdf.Google Scholar
Scanlon, T., Counting special points: Logic, Diophantine geometry, and transcendence theory. Bulletin of the American Mathematical Society, vol. 49 (2012), pp. 51–71.Google Scholar
Seidenberg, A., A new decision method for elementary algebra. Annals of Mathematics, vol. 60 (1954), pp. 365–374.Google Scholar
Seidenberg, A., Comments on Lefschetz’s principle. American American Monthly, vol. 65 (1958),pp. 685–690.Google Scholar
Shelah, S., Differentially closed fields. Israel Journal of Mathematics, vol. 16 (1973), pp. 314–328.Google Scholar
Shelah, S., Classification theory and the number of nonisomorphic models., North-Holland, Amsterdam, 1978.Google Scholar
Shelah, S., Simple unstable theories. Annals of Mathematical Logic, vol. 19 (1980), pp. 177–203.Google Scholar
Shelah, S., Classification theory for nonelementary classes. I. The number of uncountable models of ψ Є L ω 1ωpart A. Israel Journal of Mathematics, vol. 46 (1983), no. 3, pp. 212–240. paper 87a.Google Scholar
Shelah, S., Classification theory for nonelementary classes. I. The number of uncountable models of ψ Є L ω 1ωpart B. Israel Journal of Mathematics, vol. 46 (1983), no. 3, pp. 241–271. paper 87b.Google Scholar
Shelah, S., Classification theory and the number of nonisomorphic models., second edition, North-Holland, 1991.Google Scholar
Shelah, S., Toward classifying unstable theories. Annals of Pure and Applied Logic, vol. 80 (1996), pp. 229–255. paper 500.Google Scholar
Shelah, S., Classification theory for abstract elementary classes, Studies in Logic. College Publications www.collegepublications.co.uk, 2009. Binds together papers 88r, 600, 705, 734 with introduction E53.Google Scholar
Shelah, S., Response to the award of the 2013 steele prize for seminal research., A.M.S. Prize Booklet, 2013, page. 50. http://www.ams.org/profession/prizebooklet-2013.pdf.Google Scholar
Shoenfield, Joseph, Mathematical Llogic. Addison-Wesley, Reading, MA, 1967.Google Scholar
Sieg, W., Mechanical procedures and mathematical experiences., Mathematics and Mind (George, A., editor), Oxford University Press, Oxford, 1994, pp. 71–117.Google Scholar
Sieg, W., Hilbert’s programs: 1917–1922.,this Journal, vol. 5 (1999), pp. 1–44.Google Scholar
Spivak, M., Calculus., Publish or Perish Press, Houston, TX, 1980.Google Scholar
Tao, T., Infinite fields, finite fields, and the Ax-Grothendieck theorem., http://terrytao.wordpress.com/2009/03/07/infinite-fields-finite-fields-and-the-ax-grothendieck-theorem/.Google Scholar
Tarski, A. and Vaught, R.L., Arithmetical extensions of relational systems. Compositio Mathematica, vol. 13 (1956), pp. 81–102.Google Scholar
Tarski, Alfred, Sur les ensemble définissable de nombres réels I. Fundamenta Mathematica, vol. 17 (1931), pp. 210–239.Google Scholar
Teissier, B., Tame and stratified objects., Geometric Galois actions I: Around Grothendieck’s Equisse d’un Programme (Schneps, L. and Lochak, Pierre, editors), Cambridge University Press, 1997, pp. 231–242.CrossRefGoogle Scholar
Väänänen, Jouko, Second order logic or set theory, this Journal, vol. 18 (2012), pp. 91–121.Google Scholar
van den Dries, L, Macintyre, A., and Marker, D., Logarithmic-exponential power series, Journal of the London Mathematical Society, (1997), pp. 417–434.Google Scholar
Vaught, R.L., Denumerable models of complete theories., Infinitistic methods, Proceedings of the Symposium on Foundations of Mathematics, Warsaw, 1959, Państwowe Wydawnictwo Naukowe, Warsaw, 1961, pp. 303–321.Google Scholar
Veblen, Oswald, A system of axioms for geometry. Transactions of the American Mathematical Society, vol. 5 (1904), pp. 343–384, 1904.Google Scholar
Wagner, F., Relational structures and dimensions., Automorphisms of first order Sstructures (Richard, Kaye et al. ., editors), Oxford, Clarendon Press, 1994, pp 153–180.Google Scholar
Waterhouse, William, Affine group schemes., Springer-Verlag, New York, 1979.Google Scholar
Weaver, G., The model theory of Dedekind algebras, preprint, http://www.bu.edu/wcp/Papers/Logi/LogiWeav.htm.Google Scholar
Weaver, G. and George, B., The Fraenkel-Carnap question for limited higher-order languages. Bulletin of the Section of Logic, vol. 39 (2010). http://www.filozof.uni.lodz.pl/bulletin/pdf/39_12_1.pdf.Google Scholar
Weil, Andre, Foundations of algebraic geometry., American Mathematical Society, 1962. 1st edition 1946.Google Scholar
Wilkie, A., Model completeness results for expansions of the real field by restricted Pfaffian functions and exponentiation. Journal of American Mathematical Society, (1996), pp. 1051–1094.Google Scholar
Wilkie, A., O-minimal structures., Séminaire BOURBAKI, 985, 2007. http://eprints.ma.man.ac.uk/1745/01/covered/MIMS_ep2012_3.pdf.Google Scholar
Zalamea, F., Filosofía sintética de las matematáticas comtemporáneas., Colección OBRA SELECTA. Editorial Univeridad National de Colombia, Bogota, 2009.Google Scholar
Ziegler, M., Einige unentscheidbare Kĺorpertheorien. Enseignement Mathématique, vol. 28 (1982), pp. 269–280. Michael Beeson has an English translation.Google Scholar
Zil’ber, B.I., Strongly minimal countably categorical theories. Siberian Mathematics Journal, vol. 24 (1980), pp. 219–230.Google Scholar
Zil’ber, B.I., Strongly minimal countably categorical theories II. Siberian Mathematics Journal, vol. 25 (1984), pp. 396–412.Google Scholar
Zil’ber, B.I., The structure of models of uncountably categorical theories., Proceedings of the International Congress of Mathematicians August 16–23, 1983, Warszawa, pages 359–368. Polish Scientific Publishers, Warszawa, 1984. http://www.mathunion.org/ICM/ICM1983.1/Main/icm1983.1.0359.0368.ocr.pdf.Google Scholar
Zil’ber, B.I., Uncountably categorical theories., Translations of the American Mathematical Society. American Mathematical Society, 1991. summary of earlier work.Google Scholar
Zil’ber, B.I., Pseudo-exponentiation on algebraically closed fields of characteristic 0. Annals of Pure and Applied Logic, vol. 132 (2004), pp. 67–95.CrossRefGoogle Scholar
Zil’ber, B.I., A categoricity theorem for quasiminimal excellent classes, Logic and its applications, Contemporary Mathematics, Ȧmerican Mathematical Society, 2005, pp. 297–306.Google Scholar
Zil’ber, B.I., Zariski geometries: Geometry from the logicians point of view., Number 360 in London Math. Soc. Lecture Notes. London Mathematical Society, Cambridge University Press, 2010.Google Scholar