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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Logical dreams
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by Saharon Shelah PDF
Bull. Amer. Math. Soc. 40 (2003), 203-228 Request permission

Abstract:

We discuss the past and future of set theory, axiom systems and independence results. We deal in particular with cardinal arithmetic.
References
    [Baxx]BaxxTomek Bartoszyński. Invariants of Measure and Category. In M. Foreman, A. Kanamori, and M. Magidor, editors, Handbook of Set Theory. Kluwer, to appear.
  • J. Barwise and S. Feferman (eds.), Model-theoretic logics, Perspectives in Mathematical Logic, Springer-Verlag, New York, 1985. MR 819531
  • [Bsxx]BsxxAndreas Blass. Combinatorial Cardinal Characteristics of the Continuum. In M. Foreman, A. Kanamori, and M. Magidor, editors, Handbook of Set Theory. Kluwer, to appear.
  • Keith I. Devlin and R. B. Jensen, Marginalia to a theorem of Silver, $\vDash$ISILC Logic Conference (Proc. Internat. Summer Inst. and Logic Colloq., Kiel, 1974) Lecture Notes in Math., Vol. 499, Springer, Berlin, 1975, pp. 115–142. MR 0480036
  • A. Dodd and R. Jensen, The core model, Ann. Math. Logic 20 (1981), no. 1, 43–75. MR 611394, DOI 10.1016/0003-4843(81)90011-5
  • [DjSh:659]DjSh:659M. Džamonja and S. Shelah. Universal graphs at the successor of singular cardinal. J. of Symbolic Logic, to appear in 2003.
  • Paul C. Eklof and Alan H. Mekler, Almost free modules, North-Holland Mathematical Library, vol. 46, North-Holland Publishing Co., Amsterdam, 1990. Set-theoretic methods. MR 1055083
  • [EM02]EM02Paul C. Eklof and Alan Mekler. Almost free modules: Set theoretic methods, volume 65 of North Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, 2002. Revised Edition.
  • P. Erdős, A. Hajnal, and R. Rado, Partition relations for cardinal numbers, Acta Math. Acad. Sci. Hungar. 16 (1965), 93–196. MR 202613, DOI 10.1007/BF01886396
  • Fred Galvin and András Hajnal, Inequalities for cardinal powers, Ann. of Math. (2) 101 (1975), 491–498. MR 376359, DOI 10.2307/1970936
  • Moti Gitik and Saharon Shelah, More on real-valued measurable cardinals and forcing with ideals, Israel J. Math. 124 (2001), 221–242. MR 1856516, DOI 10.1007/BF02772619
  • Martin Goldstern and Saharon Shelah, Many simple cardinal invariants, Arch. Math. Logic 32 (1993), no. 3, 203–221. MR 1201650, DOI 10.1007/BF01375552
  • Thomas Jech, Set theory, Pure and Applied Mathematics, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 506523
  • Jussi Ketonen and Robert Solovay, Rapidly growing Ramsey functions, Ann. of Math. (2) 113 (1981), no. 2, 267–314. MR 607894, DOI 10.2307/2006985
  • Laurie Kirby and Jeff Paris, Accessible independence results for Peano arithmetic, Bull. London Math. Soc. 14 (1982), no. 4, 285–293. MR 663480, DOI 10.1112/blms/14.4.285
  • Menachem Magidor, Chang’s conjecture and powers of singular cardinals, J. Symbolic Logic 42 (1977), no. 2, 272–276. MR 485375, DOI 10.2307/2272130
  • [Mg1]Mg1Menachem Magidor. On the singular cardinals problem II. Annals Math., 106:517–547, 1977. [FOM]FOMStephen G. Simpson (moderator). E-mail list for discussing Foundations of Mathematics. http://www.math.psu.edu/simpson/fom/.
  • Handbook of mathematical logic, Studies in Logic and the Foundations of Mathematics, vol. 90, North-Holland Publishing Co., Amsterdam, 1977. With the cooperation of H. J. Keisler, K. Kunen, Y. N. Moschovakis and A. S. Troelstra. MR 457132
  • [Sh:E12]Sh:E12Saharon Shelah. Analytical Guide and Corrections to Cardinal Arithmetic, volume 29 of Oxford Logic Guides. Oxford University Press, 1994. math.LO/9906022.
  • Saharon Shelah, The future of set theory, Set theory of the reals (Ramat Gan, 1991) Israel Math. Conf. Proc., vol. 6, Bar-Ilan Univ., Ramat Gan, 1993, pp. 1–12. MR 1234276
  • [Sh 413]Sh:413Saharon Shelah. More Jonsson Algebras. Archive for Mathematical Logic, accepted. math.LO/9809199. [Sh 724]Sh:724Saharon Shelah. On nice equivalence relations on ${}^\lambda 2$. Archive for Mathematical Logic, accepted. math.LO/0009064. [Sh 702]Sh:702Saharon Shelah. On what I do not understand (and have something to say), model theory. Mathematica Japonica, 51:329–377, 2002. math.LO/9910158. [Sh 771]Sh:771Saharon Shelah. Polish Algebras shy from freedom. Israel Journal of Mathematics, submitted. [Sh 705]Sh:705Saharon Shelah. Toward classification theory of good $\lambda$ frames and abstract elementary classes, preprint.
  • Saharon Shelah, Proper forcing, Lecture Notes in Mathematics, vol. 940, Springer-Verlag, Berlin-New York, 1982. MR 675955, DOI 10.1007/978-3-662-21543-2
  • Saharon Shelah, On logical sentences in $\textrm {PA}$, Logic colloquium ’82 (Florence, 1982) Stud. Logic Found. Math., vol. 112, North-Holland, Amsterdam, 1984, pp. 145–160. MR 762109, DOI 10.1016/S0049-237X(08)71815-9
  • Saharon Shelah, On power of singular cardinals, Notre Dame J. Formal Logic 27 (1986), no. 2, 263–299. MR 842153, DOI 10.1305/ndjfl/1093636617
  • Saharon Shelah, Universal classes, Classification theory (Chicago, IL, 1985) Lecture Notes in Math., vol. 1292, Springer, Berlin, 1987, pp. 264–418. MR 1033033, DOI 10.1007/BFb0082242
  • Saharon Shelah, Cardinal arithmetic, Oxford Logic Guides, vol. 29, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR 1318912
  • Saharon Shelah, Proper and improper forcing, 2nd ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998. MR 1623206, DOI 10.1007/978-3-662-12831-2
  • [Sh 666]Sh:666Saharon Shelah. On what I do not understand (and have something to say). Fundamenta Mathematicae, 166:1–82, 2000. math.LO/9906113.
  • Saharon Shelah, The generalized continuum hypothesis revisited, Israel J. Math. 116 (2000), 285–321. MR 1759410, DOI 10.1007/BF02773223
  • Saharon Shelah and Lee J. Stanley, Filters, Cohen sets and consistent extensions of the Erdős-Dushnik-Miller theorem, J. Symbolic Logic 65 (2000), no. 1, 259–271. MR 1782118, DOI 10.2307/2586535
  • [Sh:E25]Sh:E25Shelah, Saharon. You Can Enter Cantor’s Paradise! In Proceedings of the Erdős Conference. Budapest’99. math.LO/0102056.
  • J. L. Walsh, On interpolation by functions analytic and bounded in a given region, Trans. Amer. Math. Soc. 46 (1939), 46–65. MR 55, DOI 10.1090/S0002-9947-1939-0000055-0
  • [Wd00]Wd00Hugh Woodin. The Continuum Hypothesis, Part I. Notices of the American Mathematical Society, 48:567–576, 2001. [Woxx]WoxxHugh Woodin. The Axiom of Determinacy, Forcing Axioms and the Nonstationary Ideal, volume 1 of de Gruyter Series in Logic and its Applications. De Gruyter, in press.
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Additional Information
  • Saharon Shelah
  • Affiliation: Institute of Mathematics, The Hebrew University, Jerusalem, Israel; Mathematics Department, Rutgers University-New Brunswick, Piscataway, New Jersey 08854-8019
  • MR Author ID: 160185
  • ORCID: 0000-0003-0462-3152
  • Email: shelah@math.huji.ac.il
  • Published electronically: February 12, 2003
  • Additional Notes: I would like to thank Alice Leonhardt for the beautiful typing. This paper is based on my lecture (and the preparations to the lecture) during the conference Mathematical Challenges of the 21st Century and is publication E23
  • © Copyright 2003 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 40 (2003), 203-228
  • MSC (2000): Primary 03-02, 03Bxx, 03Exx; Secondary 03Cxx, 03C45
  • DOI: https://doi.org/10.1090/S0273-0979-03-00981-9
  • MathSciNet review: 1962296