Book Review
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2566452
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Book Information:
Author:
Calixto Badesa
Title:
The birth of model theory: Löwenheim’s theory in the frame of the theory of relatives
Additional book information:
Princeton University Press,
Princeton, NJ,
2004,
xiv+240 pp.,
ISBN 978-0-691-05853-5,
US$64.00,
hardcover
Tuna Altınel, Alexandre V. Borovik, and Gregory Cherlin, Simple groups of finite Morley rank, Mathematical Surveys and Monographs, vol. 145, American Mathematical Society, Providence, RI, 2008. MR 2400564, DOI 10.1090/surv/145
James Ax and Simon Kochen, Diophantine problems over local fields. I, Amer. J. Math. 87 (1965), 605–630. MR 184930, DOI 10.2307/2373065
James Ax and Simon Kochen, Diophantine problems over local fields. II. A complete set of axioms for $p$-adic number theory, Amer. J. Math. 87 (1965), 631–648. MR 184931, DOI 10.2307/2373066
James Ax and Simon Kochen, Diophantine problems over local fields. III. Decidable fields, Ann. of Math. (2) 83 (1966), 437–456. MR 201378, DOI 10.2307/1970476
J. Avigad. Review of Calixto Badesa, The birth of model theory: Löwenheim’s theorem in the frame of the theory of relatives. The Mathematical Intelligencer, 28:67–71, 2006.
James Ax, The elementary theory of finite fields, Ann. of Math. (2) 88 (1968), 239–271. MR 229613, DOI 10.2307/1970573
John T. Baldwin. Categoricity. University Lecture Series, 50, American Mathematical Society, Providence, RI, 2009.
J. T. Baldwin and A. H. Lachlan, On strongly minimal sets, J. Symbolic Logic 36 (1971), 79–96. MR 286642, DOI 10.2307/2271517
L. Blum. Generalized algebraic structures: A model theoretical approach. PhD thesis, MIT, 1968.
Alexandre Borovik and Ali Nesin, Groups of finite Morley rank, Oxford Logic Guides, vol. 26, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR 1321141
Elisabeth Bouscaren (ed.), Model theory and algebraic geometry, Lecture Notes in Mathematics, vol. 1696, Springer-Verlag, Berlin, 1998. An introduction to E. Hrushovski’s proof of the geometric Mordell-Lang conjecture. MR 1678586, DOI 10.1007/978-3-540-68521-0
Zoé Chatzidakis and Ehud Hrushovski, Model theory of difference fields, Trans. Amer. Math. Soc. 351 (1999), no. 8, 2997–3071. MR 1652269, DOI 10.1090/S0002-9947-99-02498-8
G. Cherlin, L. Harrington, and A. H. Lachlan, $\aleph _0$-categorical, $\aleph _0$-stable structures, Ann. Pure Appl. Logic 28 (1985), no. 2, 103–135. MR 779159, DOI 10.1016/0168-0072(85)90023-5
Raf Cluckers and François Loeser, Constructible motivic functions and motivic integration, Invent. Math. 173 (2008), no. 1, 23–121. MR 2403394, DOI 10.1007/s00222-008-0114-1
L. Van den Dries. Tame topology and $o$-minimal structures. London Mathematical Society Lecture Note Series, 248, 1999.
J. Denef, The rationality of the Poincaré series associated to the $p$-adic points on a variety, Invent. Math. 77 (1984), no. 1, 1–23. MR 751129, DOI 10.1007/BF01389133
J. Denef and F. Loeser, Motivic integration and the Grothendieck group of pseudo-finite fields, Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002) Higher Ed. Press, Beijing, 2002, pp. 13–23. MR 1957016
Yu Ershov. Ob elementarni teorii maksimal’nykh normirovannykh polei (on the elementary theory of maximal normed fields) i. Algebra ui Logika, 4:31–69, 1965.
David M. Evans, Homogeneous geometries, Proc. London Math. Soc. (3) 52 (1986), no. 2, 305–327. MR 818929, DOI 10.1112/plms/s3-52.2.305
K. Gödel. Uber die vollständigkeit des logikkalküls. In S. Feferman et. al., editor, Kurt Gödel: Collected Works, vol. 1, pages 60–101. Oxford University Press, New York, 1929. 1929 PhD. thesis reprinted.
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 255. MR 217086
Ehud Hrushovski, Computing the Galois group of a linear differential equation, Differential Galois theory (Będlewo, 2001) Banach Center Publ., vol. 58, Polish Acad. Sci. Inst. Math., Warsaw, 2002, pp. 97–138. MR 1972449, DOI 10.4064/bc58-0-9
Deirdre Haskell, Ehud Hrushovski, and Dugald Macpherson, Stable domination and independence in algebraically closed valued fields, Lecture Notes in Logic, vol. 30, Association for Symbolic Logic, Chicago, IL; Cambridge University Press, Cambridge, 2008. MR 2369946
Ehud Hrushovski and David Kashdan. Integration in valued fields. math ArXiv, 2xxx.
Ehud Hrushovski, Unidimensional theories are superstable, Ann. Pure Appl. Logic 50 (1990), no. 2, 117–138. MR 1081816, DOI 10.1016/0168-0072(90)90046-5
Ehud Hrushovski, The Mordell-Lang conjecture for function fields, J. Amer. Math. Soc. 9 (1996), no. 3, 667–690. MR 1333294, DOI 10.1090/S0894-0347-96-00202-0
A. Joyal. Le théorème de Chevalley-Tarski. Cah. Topol. Géom. Différ., 16:256–258, 1975.
Byunghan Kim and Anand Pillay, Simple theories, Ann. Pure Appl. Logic 88 (1997), no. 2-3, 149–164. Joint AILA-KGS Model Theory Meeting (Florence, 1995). MR 1600895, DOI 10.1016/S0168-0072(97)00019-5
L. Löwenheim. On possibilities in the calculus of relatives. In J. Van Heijenoort, editor, From Frege to Godel: A Sourcebook in Mathematical Logic, 1879-1931. Harvard University Press, 1967. German original published in 1915.
Angus Macintyre, On definable subsets of $p$-adic fields, J. Symbolic Logic 41 (1976), no. 3, 605–610. MR 485335, DOI 10.2307/2272038
Anatoliĭ Ivanovič Mal′cev, The metamathematics of algebraic systems. Collected papers: 1936–1967, Studies in Logic and the Foundations of Mathematics, Vol. 66, North-Holland Publishing Co., Amsterdam-London, 1971. Translated, edited, and provided with supplementary notes by Benjamin Franklin Wells, III. MR 0349383
David Marker, The number of countable differentially closed fields, Notre Dame J. Formal Logic 48 (2007), no. 1, 99–113. MR 2289900, DOI 10.1305/ndjfl/1172787548
Michael Morley, Categoricity in power, Trans. Amer. Math. Soc. 114 (1965), 514–538. MR 175782, DOI 10.1090/S0002-9947-1965-0175782-0
Michael Makkai and Gonzalo E. Reyes, First order categorical logic, Lecture Notes in Mathematics, Vol. 611, Springer-Verlag, Berlin-New York, 1977. Model-theoretical methods in the theory of topoi and related categories. MR 0505486
Anand Pillay and Charles Steinhorn, Definable sets in ordered structures. I, Trans. Amer. Math. Soc. 295 (1986), no. 2, 565–592. MR 833697, DOI 10.1090/S0002-9947-1986-0833697-X
Anand Pillay, Book Review: Uncountably categorical theories, Bull. Amer. Math. Soc. (N.S.) 30 (1994), no. 2, 302–308. MR 1568097, DOI 10.1090/S0273-0979-1994-00473-2
Anand Pillay, Geometric stability theory, Oxford Logic Guides, vol. 32, The Clarendon Press, Oxford University Press, New York, 1996. Oxford Science Publications. MR 1429864
Bruno Poizat, Une théorie de Galois imaginaire, J. Symbolic Logic 48 (1983), no. 4, 1151–1170 (1984) (French). MR 727805, DOI 10.2307/2273680
Bruno Poizat, Groupes stables, Nur al-Mantiq wal-Maʾrifah [Light of Logic and Knowledge], vol. 2, Bruno Poizat, Lyon, 1987 (French). Une tentative de conciliation entre la géométrie algébrique et la logique mathématique. [An attempt at reconciling algebraic geometry and mathematical logic]. MR 902156
M. Pressburger. Uber die vollstandigkeit eines gewissen Systems der Arithmetic ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. In Sprawozdanie z I Kongresu Mat. Karjów Słowiańskich, pages 92–101, 1930.
Abraham Robinson, On predicates in algebraically closed fields, J. Symbolic Logic 19 (1954), 103–114. MR 62090, DOI 10.2307/2268866
Abraham Robinson, On the concept of a differentially closed field, Bull. Res. Council Israel Sect. F 8F (1959), 113–128 (1959). MR 125016
Gerald E. Sacks, The differential closure of a differential field, Bull. Amer. Math. Soc. 78 (1972), 629–634. MR 299466, DOI 10.1090/S0002-9904-1972-12969-0
Saharon Shelah, Stability, the f.c.p., and superstability; model theoretic properties of formulas in first order theory, Ann. Math. Logic 3 (1971), no. 3, 271–362. MR 317926, DOI 10.1016/0003-4843(71)90015-5
Saharon Shelah, Differentially closed fields, Israel J. Math. 16 (1973), 314–328. MR 344116, DOI 10.1007/BF02756711
Saharon Shelah, The lazy model-theoretician’s guide to stability, Logique et Anal. (N.S.) 18 (1975), no. 71-72, 241–308. MR 539969
Saharon Shelah, Simple unstable theories, Ann. Math. Logic 19 (1980), no. 3, 177–203. MR 595012, DOI 10.1016/0003-4843(80)90009-1
Saharon Shelah, Classification theory for nonelementary classes. I. The number of uncountable models of $\psi \in L_{\omega _{1},\omega }$. Part A, Israel J. Math. 46 (1983), no. 3, 212–240. MR 733351, DOI 10.1007/BF02761954
Saharon Shelah, Classification theory for nonelementary classes. I. The number of uncountable models of $\psi \in L_{\omega _{1},\omega }$. Part B, Israel J. Math. 46 (1983), no. 4, 241–273. MR 730343, DOI 10.1007/BF02762887
S. Shelah, Classification theory and the number of nonisomorphic models, 2nd ed., Studies in Logic and the Foundations of Mathematics, vol. 92, North-Holland Publishing Co., Amsterdam, 1990. MR 1083551
S. Shelah. Classification theory for abstract elementary classes. Studies in Logic, College Publications, www.collegepublications.co.uk (binds together papers 88r, 300, 600, 705, 734, 838 with introduction E53), 2009.
T. Tao. 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
Alfred Tarski. Sur les ensemble définissable de nombres réels I. Fundamenta Mathematica, 17:210–239, 1931.
Alfred Tarski, A decision method for elementary algebra and geometry, University of California Press, Berkeley-Los Angeles, Calif., 1951. 2nd ed. MR 0044472
Alfred Tarski and Robert L. Vaught, Arithmetical extensions of relational systems, Compositio Math. 13 (1958), 81–102. MR 95121
Lou van den Dries, Angus Macintyre, and David Marker, Logarithmic-exponential power series, J. London Math. Soc. (2) 56 (1997), no. 3, 417–434. MR 1610431, DOI 10.1112/S0024610797005437
Jean van Heijenoort, From Frege to Gödel. A source book in mathematical logic, 1879–1931, Harvard University Press, Cambridge, Mass., 1967. MR 0209111
A. J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted Pfaffian functions and the exponential function, J. Amer. Math. Soc. 9 (1996), no. 4, 1051–1094. MR 1398816, DOI 10.1090/S0894-0347-96-00216-0
B.I. Zil’ber. The structure of models of uncountably categorical theories. Polish Scientific Publishers, Warszawa, 1984.
B. Zilber, Model theory, geometry and arithmetic of the universal cover of a semi-abelian variety, Model theory and applications, Quad. Mat., vol. 11, Aracne, Rome, 2002, pp. 427–458. MR 2159728
B.I. Zilber. Pseudo-exponentiation on algebraically closed fields of characteristic 0. Annals of Pure and Applied Logic, 132:67–95, 2004.
B.I. Zilber. A categoricity theorem for quasiminimal excellent classes. In Logic and its Applications, Contemporary Mathematics, pages 297–306. AMS, 2005.
B.I. Zilber. Covers of the multiplicative group of an algebraically closed field of characteristic 0. J. London Math. Soc., pages 41–58, 2006.
References
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- J. Ax and S. Kochen. Diophantine problems over local fields II: A complete set of axioms for $p$-adic number theory. Amer. J. of Math., 87:631–48, 1965. MR 0184931 (32:2402)
- J. Ax and S. Kochen. Diophantine problems over local fields III: Decidable fields. Annals of Math, 83:437–456, 1966. MR 0201378 (34:1262)
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- James Ax. The elementary theory of finite fields. Annals of Math, 88:239–271, 1968. MR 0229613 (37:5187)
- John T. Baldwin. Categoricity. University Lecture Series, 50, American Mathematical Society, Providence, RI, 2009.
- J.T. Baldwin and A.H. Lachlan. On strongly minimal sets. Journal of Symbolic Logic, 36:79–96, 1971. MR 0286642 (44:3851)
- L. Blum. Generalized algebraic structures: A model theoretical approach. PhD thesis, MIT, 1968.
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- E. Bouscaren, editor. Model theory and algebraic geometry: An introduction to E. Hrushovski’s proof of the geometric Mordell-Lang conjecture. Springer-Verlag, 1999. MR 1678586 (99k:03032)
- Z. Chatzidakis and U. Hrushovski. The model theory of difference fields. Transactions of the AMS, 351:2997–3071, 1999. MR 1652269 (2000f:03109)
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- R. Cluckers and F. Loeser. Constructible motivic functions and motivic integration. Inventiones Mathematicae, 173:23–121, 2008. MR 2403394 (2009g:14018)
- L. Van den Dries. Tame topology and $o$-minimal structures. London Mathematical Society Lecture Note Series, 248, 1999.
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- D. Evans. Homogeneous geometries. Proc. London Math. Soc., 52:305–327, 1986. MR 818929 (87f:51018)
- K. Gödel. Uber die vollständigkeit des logikkalküls. In S. Feferman et. al., editor, Kurt Gödel: Collected Works, vol. 1, pages 60–101. Oxford University Press, New York, 1929. 1929 PhD. thesis reprinted.
- A Grothendieck. Elements de géometrie algébriques (rédigés avec la collaboration de J. Dieudonné), IV, Etudes Locale des Schémas et des Morphismes de Schémas. Publ Math I.H.E.S., 28:5–255, 1966. MR 0217086 (36:178)
- E. Hrushovski, “Computing the Galois group of a linear differential equation”, in Differential Galois Theory (Bedlewo, 2001), 97–138, Banach Center Publ., 58, Polish Acad. Sci., Warsaw, 2002. MR 1972449 (2004j:12007)
- D. Haskell, E. Hrushovski, and H.D. MacPherson. Stable domination and independence in algebraically closed valued fields. Lecture Notes in Logic. Association for Symbolic Logic, 2007. MR 2369946
- Ehud Hrushovski and David Kashdan. Integration in valued fields. math ArXiv, 2xxx.
- E. Hrushovski. Unidimensional theories are superstable. Annals of Pure and Applied Logic, 50:117–138, 1990. MR 1081816 (92g:03052)
- E. Hrushovski. The Mordell-Lang conjecture over function fields. Journal of the American Mathematical Society, 9:667–690, 1996. MR 1333294 (97h:11154)
- A. Joyal. Le théorème de Chevalley-Tarski. Cah. Topol. Géom. Différ., 16:256–258, 1975.
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- L. Löwenheim. On possibilities in the calculus of relatives. In J. Van Heijenoort, editor, From Frege to Godel: A Sourcebook in Mathematical Logic, 1879-1931. Harvard University Press, 1967. German original published in 1915.
- Angus J. Macintyre. On definable subsets of $p$-adic fields. Journal of Symbolic Logic, 41:605–10, 1976. MR 0485335 (58:5182)
- A. I. Malcev. “Investigations in the area of mathematical logic”, in The metamathematics of algebraic systems, Collected papers: 1936–1967 (B. F. Wells, editor), Studies in Mathematics and the Foundations of Mathematics, vol. 66. North-Holland, Amsterdam, pp. 1–14. German original published in Mat. Sbornik, 1936. MR 0349383 (50:1877)
- D. Marker. The number of countable differentially closed fields. Notre Journal of Formal Logic, 48:99–113, 2007. MR 2289900 (2007m:03076)
- M. Morley. Categoricity in power. Transactions of the AMS, 114:514–538, 1965. MR 0175782 (31:58)
- M. Makkai and G.E. Reyes. First order categorical logic, volume 111 of Springer Lecture Notes. Springer-Verlag, 1977. MR 0505486 (58:21600)
- A. Pillay and Steinhorn C. Definable sets in ordered structures I. Transactions of the AMS, 295, 1986. MR 833697 (88b:03050a)
- A. Pillay. Review of “Uncountably categorical theories”, by Boris Zilber. Bulletin of the AMS, 30:302–308, 1994. MR 1568097
- A. Pillay. Geometric stability theory. Clarendon Press, Oxford, 1996. MR 1429864 (98a:03049)
- Bruno Poizat. Une théorie de Galois imaginaire. Journal of Symbolic Logic, pages 1151–1170, 1983. MR 727805 (85e:03083)
- Bruno Poizat. Groupes Stables. Nur Al-mantiq Wal-ma’rifah, 82, Rue Racine 69100 Villeurbanne France, 1987. MR 902156 (89b:03056)
- M. Pressburger. Uber die vollstandigkeit eines gewissen Systems der Arithmetic ganzer Zahlen, in welchem die Addition als einzige Operation hervortritt. In Sprawozdanie z I Kongresu Mat. Karjów Słowiańskich, pages 92–101, 1930.
- A. Robinson. On predicates in algebraically closed fields. The Journal of Symbolic Logic, 19:103–114, 1954. MR 0062090 (15:925f)
- A. Robinson. On the concept of a differentially closed field. Bull. Res. Council Israel, 8F:113–128, 1959. MR 0125016 (23:A2323)
- Gerald E. Sacks. The differential closure of a differential field. Bulletin of the AMS, 78:629–634, 1972. MR 0299466 (45:8514)
- S. Shelah. Stability, the f.c.p., and superstability; model theoretic properties of formulas in first-order theories. Annals of Mathematical Logic, 3:271–362, 1970. MR 0317926 (47:6475)
- S. Shelah. Differentially closed fields. Israel Journal of Mathematics, 16:314–328, 1973. MR 0344116 (49:8856)
- S. Shelah. The lazy model-theoretician’s guide to stability. Logique et Analyse, 18:241–308, 1975. MR 0539969 (58:27447)
- Saharon Shelah. Simple unstable theories. Annals of Mathematical Logic, 19:177–203, 1980. MR 595012 (82g:03055)
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- S. Shelah. Classification theory for abstract elementary classes. Studies in Logic, College Publications, www.collegepublications.co.uk (binds together papers 88r, 300, 600, 705, 734, 838 with introduction E53), 2009.
- T. Tao. 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
- Alfred Tarski. Sur les ensemble définissable de nombres réels I. Fundamenta Mathematica, 17:210–239, 1931.
- Alfred Tarski. A decision method for elementary algebra and geometry. University of California Press, Berkeley and Los Angeles, Calif., 1951. 2nd edition, . MR 0044472 (13:423a)
- A. Tarski and R.L. Vaught. Arithmetical extensions of relational systems. Compositio Mathematica, 13:81–102, 1956. MR 0095121 (20:1627)
- L. van den Dries, A. Macintyre, and D. Marker. Logarithmic-exponential power series. J. London Math. Soc., pages 417–434, 1997. MR 1610431 (99d:03063)
- J. Van Heijenoort, editor. From Frege to Godel: A sourcebook in mathematical logic, 1879-1931. Harvard University Press, 1967. MR 0209111 (35:15)
- A. Wilkie. Model completeness results for expansions of the real field by restricted pfaffian functions and exponentiation. Journal of the AMS, pages 1051–1094, 1996. MR 1398816 (98j:03052)
- B.I. Zil’ber. The structure of models of uncountably categorical theories. Polish Scientific Publishers, Warszawa, 1984.
- B.I. Zilber. Model theory, geometry and arithmetic of universal covers of a semi-abelian variety. In Model Theory and Applications, Quaterna di matematica, pages 427–458, 2003. MR 2159728 (2006g:03067)
- B.I. Zilber. Pseudo-exponentiation on algebraically closed fields of characteristic 0. Annals of Pure and Applied Logic, 132:67–95, 2004.
- B.I. Zilber. A categoricity theorem for quasiminimal excellent classes. In Logic and its Applications, Contemporary Mathematics, pages 297–306. AMS, 2005.
- B.I. Zilber. Covers of the multiplicative group of an algebraically closed field of characteristic 0. J. London Math. Soc., pages 41–58, 2006.
Review Information:
Reviewer:
John T. Baldwin
Affiliation:
University of Illinois at Chicago
Journal:
Bull. Amer. Math. Soc.
47 (2010), 177-185
DOI:
https://doi.org/10.1090/S0273-0979-09-01275-0
Published electronically:
September 8, 2009
Additional Notes:
The author was partially supported by NSF grant DMS-0500841.
Review copyright:
© Copyright 2009
American Mathematical Society