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Book Information
Author(s):
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.,
(hardcover)
US$64.00,
ISBN 978-0-691-05853-5
References:
-
- [ABC08]
- T. Altinel, A. Borovik, and G. Cherlin.
Simple Groups of Finite Morley Rank of Even Type. Mathematical Surveys and Monographs, 145. American Mathematical Society, 2008. MR 2400564 (2009a:20046) - [AK65a]
- J. Ax and S. Kochen.
Diophantine problems over local fields I:. Amer. J. of Math., 87:605-30, 1965. MR 0184930 (32:2401) - [AK65b]
- J. Ax and S. Kochen.
Diophantine problems over local fields II: A complete set of axioms for -adic number theory. Amer. J. of Math., 87:631-48, 1965. MR 0184931 (32:2402) - [AK66]
- J. Ax and S. Kochen.
Diophantine problems over local fields III: Decidable fields. Annals of Math, 83:437-456, 1966. MR 0201378 (34:1262) - [Avi06]
- 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. - [Ax68]
- James Ax.
The elementary theory of finite fields. Annals of Math, 88:239-271, 1968. MR 0229613 (37:5187) - [Bal]
- John T. Baldwin.
Categoricity. University Lecture Series, 50, American Mathematical Society, Providence, RI, 2009. - [BL71]
- J.T. Baldwin and A.H. Lachlan.
On strongly minimal sets. Journal of Symbolic Logic, 36:79-96, 1971. MR 0286642 (44:3851) - [Blu68]
- L. Blum.
Generalized algebraic structures: A model theoretical approach. PhD thesis, MIT, 1968. - [BN94]
- A. Borovik and A. Nesin.
Groups of finite Morley rank. Oxford University Press, 1994. MR 1321141 (96c:20004) - [Bou99]
- 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) - [CH99]
- Z. Chatzidakis and U. Hrushovski.
The model theory of difference fields. Transactions of the AMS, 351:2997-3071, 1999. MR 1652269 (2000f:03109) - [CHL85]
- G.L. Cherlin, L. Harrington, and A.H. Lachlan.
-categorical, -stable structures. Annals of Pure and Applied Logic, 28:103-135, 1985. MR 779159 (86g:03054) - [CL08]
- R. Cluckers and F. Loeser.
Constructible motivic functions and motivic integration. Inventiones Mathematicae, 173:23-121, 2008. MR 2403394 (2009g:14018) - [dD99]
- L. Van den Dries.
Tame topology and -minimal structures. London Mathematical Society Lecture Note Series, 248, 1999. - [Den84]
- J. Denef.
Rationality of the Poincarè series associated to the -adic points on a variety. Inventiones Mathematica, 72:1-23, 1984. MR 751129 (86c:11043) - [DL02]
- J. Denef and F. Loeser.
Motivic integration and the Grothendieck group of pseudo-finite fields. In Proceedings of the International Congress of Mathematicians, Beijing 2002,, volume 2, pages 13-23. Higher Education Press, Beijing, 2002. MR 1957016 (2004f:14040) - [Ers65]
- 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. - [Eva86]
- D. Evans.
Homogeneous geometries. Proc. London Math. Soc., 52:305-327, 1986. MR 818929 (87f:51018) - [Göd29]
- 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. - [Gro66]
- 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) - [Hru02]
- 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)
- [HHM07]
- 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 - [HKxx]
- Ehud Hrushovski and David Kashdan.
Integration in valued fields. math ArXiv, 2xxx. - [Hru90]
- E. Hrushovski.
Unidimensional theories are superstable. Annals of Pure and Applied Logic, 50:117-138, 1990. MR 1081816 (92g:03052) - [Hru96]
- E. Hrushovski.
The Mordell-Lang conjecture over function fields. Journal of the American Mathematical Society, 9:667-690, 1996. MR 1333294 (97h:11154) - [Joy75]
- A. Joyal.
Le théorème de Chevalley-Tarski. Cah. Topol. Géom. Différ., 16:256-258, 1975. - [KP97]
- B. Kim and A. Pillay.
Simple theories. Annals of Pure and Applied Logic, 88:149-164, 1997. MR 1600895 (99b:03049) - [Löw67]
- 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. - [Mac76]
- Angus J. Macintyre.
On definable subsets of -adic fields. Journal of Symbolic Logic, 41:605-10, 1976. MR 0485335 (58:5182) - [Mal72]
- 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) - [Mar07]
- D. Marker.
The number of countable differentially closed fields. Notre Journal of Formal Logic, 48:99-113, 2007. MR 2289900 (2007m:03076) - [Mor65]
- M. Morley.
Categoricity in power. Transactions of the AMS, 114:514-538, 1965. MR 0175782 (31:58) - [MR77]
- M. Makkai and G.E. Reyes.
First order categorical logic, volume 111 of Springer Lecture Notes. Springer-Verlag, 1977. MR 0505486 (58:21600) - [PC86]
- A. Pillay and Steinhorn C.
Definable sets in ordered structures I. Transactions of the AMS, 295, 1986. MR 833697 (88b:03050a) - [Pil94]
- A. Pillay.
Review of ``Uncountably categorical theories'', by Boris Zilber. Bulletin of the AMS, 30:302-308, 1994. MR 1568097 - [Pil96]
- A. Pillay.
Geometric stability theory. Clarendon Press, Oxford, 1996. MR 1429864 (98a:03049) - [Poi83]
- Bruno Poizat.
Une théorie de Galois imaginaire. Journal of Symbolic Logic, pages 1151-1170, 1983. MR 727805 (85e:03083) - [Poi87]
- Bruno Poizat.
Groupes Stables. Nur Al-mantiq Wal-ma'rifah, 82, Rue Racine 69100 Villeurbanne France, 1987. MR 902156 (89b:03056) - [Pre30]
- 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. - [Rob54]
- A. Robinson.
On predicates in algebraically closed fields. The Journal of Symbolic Logic, 19:103-114, 1954. MR 0062090 (15:925f) - [Rob59]
- A. Robinson.
On the concept of a differentially closed field. Bull. Res. Council Israel, 8F:113-128, 1959. MR 0125016 (23:A2323) - [Sac72]
- Gerald E. Sacks.
The differential closure of a differential field. Bulletin of the AMS, 78:629-634, 1972. MR 0299466 (45:8514) - [She70]
- 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) - [She73]
- S. Shelah.
Differentially closed fields. Israel Journal of Mathematics, 16:314-328, 1973. MR 0344116 (49:8856) - [She75]
- S. Shelah.
The lazy model-theoretician's guide to stability. Logique et Analyse, 18:241-308, 1975. MR 0539969 (58:27447) - [She80]
- Saharon Shelah.
Simple unstable theories. Annals of Mathematical Logic, 19:177-203, 1980. MR 595012 (82g:03055) - [She83a]
- S. Shelah.
Classification theory for nonelementary classes. I. The number of uncountable models of part A. Israel Journal of Mathematics, 46:3:212-240, 1983. paper 87a. MR 733351 (85m:03024a) - [She83b]
- S. Shelah.
Classification theory for nonelementary classes. I. The number of uncountable models of part B. Israel Journal of Mathematics, 46;3:241-271, 1983. paper 87b. MR 730343 (85m:03024b) - [She91]
- S. Shelah.
Classification theory and the number of nonisomorphic models. North-Holland, 1991. Second Edition. MR 1083551 (91k:03085) - [She00]
- 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. - [Tao]
- 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 - [Tar31]
- Alfred Tarski.
Sur les ensemble définissable de nombres réels I. Fundamenta Mathematica, 17:210-239, 1931. - [Tar51]
- Alfred Tarski.
A decision method for elementary algebra and geometry. University of California Press, Berkeley and Los Angeles, Calif., 1951. 2nd edition, <http://www.ams.org/mathscinet/se arch/publications.html?pg1=IID&s1=170920>. MR 0044472 (13:423a) - [TV56]
- A. Tarski and R.L. Vaught.
Arithmetical extensions of relational systems. Compositio Mathematica, 13:81-102, 1956. MR 0095121 (20:1627) - [vdDMM97]
- 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) - [VH67]
- J. Van Heijenoort, editor.
From Frege to Godel: A sourcebook in mathematical logic, 1879-1931. Harvard University Press, 1967. MR 0209111 (35:15) - [Wil96]
- 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) - [Zil84]
- B.I. Zil'ber.
The structure of models of uncountably categorical theories. Polish Scientific Publishers, Warszawa, 1984. - [Zil03]
- 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) - [Zil04]
- B.I. Zilber.
Pseudo-exponentiation on algebraically closed fields of characteristic 0. Annals of Pure and Applied Logic, 132:67-95, 2004. - [Zil05]
- B.I. Zilber.
A categoricity theorem for quasiminimal excellent classes. In Logic and its Applications, Contemporary Mathematics, pages 297-306. AMS, 2005. - [Zil06]
- B.I. Zilber.
Covers of the multiplicative group of an algebraically closed field of characteristic 0. J. London Math. Soc., pages 41-58, 2006.
Additional Information:
Reviewer(s):
John
T.
Baldwin
Affiliation:
University of Illinois at Chicago
Review Information:
Journal:
Bull. Amer. Math. Soc.
47
(2010),
177-185.
MSC
(2000):
Primary 01A60, 03C07
DOI:
10.1090/S0273-0979-09-01275-0
PII:
S 0273-0979(09)01275-0
Posted:
September 8, 2009
Additional notes:
The author was partially supported by NSF grant DMS-0500841.
Copyright of article:
Copyright
2009,
American Mathematical Society
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