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Logical dreams
Author:
Saharon Shelah
Journal:
Bull. Amer. Math. Soc. 40 (2003), 203-228
MSC (2000):
Primary 03-02, 03Bxx, 03Exx; Secondary 03Cxx, 03C45
Posted:
February 12, 2003
MathSciNet review:
1962296
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Similar Articles |
Additional Information
Abstract: We discuss the past and future of set theory, axiom systems and independence results. We deal in particular with cardinal arithmetic.
- [Baxx]
Tomek Bartoszynski.
Invariants of Measure and Category. In M. Foreman, A. Kanamori, and M. Magidor, editors, Handbook of Set Theory. Kluwer, to appear.
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J.
Barwise and S.
Feferman (eds.), Model-theoretic logics, Perspectives in
Mathematical Logic, Springer-Verlag, New York, 1985. MR 819531
(87g:03033)
- [Bsxx]
Andreas Blass.
Combinatorial Cardinal Characteristics of the Continuum. In M. Foreman, A. Kanamori, and M. Magidor, editors, Handbook of Set Theory. Kluwer, to appear.
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Keith
I. Devlin and R.
B. Jensen, Marginalia to a theorem of Silver, ⊨ISILC
Logic Conference (Proc. Internat. Summer Inst. and Logic Colloq., Kiel,
1974), Springer, Berlin, 1975, pp. 115–142. Lecture Notes in
Math., Vol. 499. MR 0480036
(58 #235)
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A.
Dodd and R.
Jensen, The core model, Ann. Math. Logic 20
(1981), no. 1, 43–75. MR 611394
(82i:03063), http://dx.doi.org/10.1016/0003-4843(81)90011-5
- [DjSh:659]
M. Dzamonja and S. Shelah.
Universal graphs at the successor of singular cardinal. J. of Symbolic Logic, to appear in 2003.
- [EM]
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
(92e:20001)
- [EM02]
Paul 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.
- [EHR]
P.
Erdős, A.
Hajnal, and R.
Rado, Partition relations for cardinal numbers, Acta Math.
Acad. Sci. Hungar. 16 (1965), 93–196. MR 0202613
(34 #2475)
- [GH]
Fred
Galvin and András
Hajnal, Inequalities for cardinal powers, Ann. of Math. (2)
101 (1975), 491–498. MR 0376359
(51 #12535)
- [GiSh 582]
Moti
Gitik and Saharon
Shelah, More on real-valued measurable cardinals and forcing with
ideals, Israel J. Math. 124 (2001), 221–242. MR 1856516
(2002g:03110), http://dx.doi.org/10.1007/BF02772619
- [GoSh 448]
Martin
Goldstern and Saharon
Shelah, Many simple cardinal invariants, Arch. Math. Logic
32 (1993), no. 3, 203–221. MR 1201650
(94c:03064), http://dx.doi.org/10.1007/BF01375552
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Thomas
Jech, Set theory, Academic Press [Harcourt Brace Jovanovich
Publishers], New York, 1978. Pure and Applied Mathematics. MR 506523
(80a:03062)
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Jussi
Ketonen and Robert
Solovay, Rapidly growing Ramsey functions, Ann. of Math. (2)
113 (1981), no. 2, 267–314. MR 607894
(84c:03100), http://dx.doi.org/10.2307/2006985
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Laurie
Kirby and Jeff
Paris, Accessible independence results for Peano arithmetic,
Bull. London Math. Soc. 14 (1982), no. 4,
285–293. MR
663480 (83j:03096), http://dx.doi.org/10.1112/blms/14.4.285
- [Mg2]
Menachem
Magidor, Chang’s conjecture and powers of singular
cardinals, J. Symbolic Logic 42 (1977), no. 2,
272–276. MR 0485375
(58 #5221)
- [Mg1]
Menachem Magidor.
On the singular cardinals problem II. Annals Math., 106:517-547, 1977.
- [FOM]
Stephen G. Simpson (moderator).
E-mail list for discussing Foundations of Mathematics. http://www.math.psu.edu/simpson/fom/.
- [PH77]
Handbook of mathematical logic, North-Holland Publishing Co.,
Amsterdam, 1977. Edited by Jon Barwise; With the cooperation of H. J.
Keisler, K. Kunen, Y. N. Moschovakis and A. S. Troelstra; Studies in Logic
and the Foundations of Mathematics, Vol. 90. MR 0457132
(56 #15351)
- [Sh:E12]
Saharon Shelah.
Analytical Guide and Corrections to []. math.LO/9906022.
- [Sh:E16]
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
(94i:03002)
- [Sh 413]
Saharon Shelah.
More Jonsson Algebras. Archive for Mathematical Logic, accepted. math.LO/9809199.
- [Sh 724]
Saharon Shelah.
On nice equivalence relations on . Archive for Mathematical Logic, accepted. math.LO/0009064.
- [Sh 702]
Saharon 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]
Saharon Shelah.
Polish Algebras shy from freedom. Israel Journal of Mathematics, submitted.
- [Sh 705]
Saharon Shelah.
Toward classification theory of good frames and abstract elementary classes, preprint.
- [Sh:b]
Saharon
Shelah, Proper forcing, Lecture Notes in Mathematics,
vol. 940, Springer-Verlag, Berlin, 1982. MR 675955
(84h:03002)
- [Sh 170]
Saharon
Shelah, On logical sentences in 𝑃𝐴, Logic
colloquium ’82 (Florence, 1982) Stud. Logic Found. Math.,
vol. 112, North-Holland, Amsterdam, 1984, pp. 145–160. MR 762109
(86g:03097), http://dx.doi.org/10.1016/S0049-237X(08)71815-9
- [Sh 111]
Saharon
Shelah, On power of singular cardinals, Notre Dame J. Formal
Logic 27 (1986), no. 2, 263–299. MR 842153
(87j:04006), http://dx.doi.org/10.1305/ndjfl/1093636617
- [Sh 300]
Saharon
Shelah, Universal classes, Classification theory (Chicago, IL,
1985) Lecture Notes in Math., vol. 1292, Springer, Berlin, 1987,
pp. 264–418. MR 1033033
(91k:03088), http://dx.doi.org/10.1007/BFb0082242
- [Sh:g]
Saharon
Shelah, Cardinal arithmetic, Oxford Logic Guides,
vol. 29, The Clarendon Press Oxford University Press, New York, 1994.
Oxford Science Publications. MR 1318912
(96e:03001)
- [Sh:f]
Saharon
Shelah, Proper and improper forcing, 2nd ed., Perspectives in
Mathematical Logic, Springer-Verlag, Berlin, 1998. MR 1623206
(98m:03002)
- [Sh 666]
Saharon Shelah.
On what I do not understand (and have something to say). Fundamenta Mathematicae, 166:1-82, 2000. math.LO/9906113.
- [Sh 460]
Saharon
Shelah, The generalized continuum hypothesis revisited, Israel
J. Math. 116 (2000), 285–321. MR 1759410
(2001g:03095), http://dx.doi.org/10.1007/BF02773223
- [ShSt 419]
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
(2001j:03085), http://dx.doi.org/10.2307/2586535
- [Sh:E25]
Shelah, Saharon.
You Can Enter Cantor's Paradise! In Proceedings of the Erdos Conference. Budapest'99. math.LO/0102056.
- [Si]
Jack
Silver, On the singular cardinals problem, Proceedings of the
International Congress of Mathematicians (Vancouver, B. C., 1974), Vol. 1,
Canad. Math. Congress, Montreal, Que., 1975, pp. 265–268. MR 0429564
(55 #2576)
- [Wd00]
Hugh Woodin.
The Continuum Hypothesis, Part I. Notices of the American Mathematical Society, 48:567-576, 2001.
- [Woxx]
Hugh 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.
- [Baxx]
- Tomek Bartoszynski.
Invariants of Measure and Category. In M. Foreman, A. Kanamori, and M. Magidor, editors, Handbook of Set Theory. Kluwer, to appear.
- [BF]
- Jon Barwise and Solomon Feferman (editors).
Model-theoretic logics. Springer Verlag, Heidelberg-New York, 1985. MR 87g:03033
- [Bsxx]
- Andreas Blass.
Combinatorial Cardinal Characteristics of the Continuum. In M. Foreman, A. Kanamori, and M. Magidor, editors, Handbook of Set Theory. Kluwer, to appear.
- [DeJ]
- Keith J. Devlin and Ronald B. Jensen.
Marginalia to a theorem of Silver. In G. H. Müller, A. Oberschelp, and K. Potthoff, editors, Proceedings of the Logic Colloquium Kiel 1974, volume 499 of Lecture Notes in Mathematicas, pages 115-142, Berlin, 1975. Springer. MR 58:235
- [DJ1]
- A. Dodd and Ronald B. Jensen.
The Core model. Annals of Math Logic, 20:43-75, 1981. MR 82i:03063
- [DjSh:659]
- M. Dzamonja and S. Shelah.
Universal graphs at the successor of singular cardinal. J. of Symbolic Logic, to appear in 2003.
- [EM]
- Paul C. Eklof and Alan Mekler.
Almost free modules: Set theoretic methods, volume 46 of North Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, 1990. MR 92e:20001
- [EM02]
- Paul 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.
- [EHR]
- Paul Erdos, Andras Hajnal, and Richard Rado.
Partition relations for cardinal numbers. Acta Math. Acad. Sci. Hung., 16:93-196, 1965. MR 34:2475
- [GH]
- Fred Galvin and Andras Hajnal.
Inequalities for cardinal powers. Annals Math., 101:491-498, 1975. MR 51:12535
- [GiSh 582]
- Moti Gitik and Saharon Shelah.
More on real-valued measurable cardinals and forcing with ideals. Israel Journal of Mathematics, 124:221-242, 2001. math.LO/9507208. MR 2002g:03110
- [GoSh 448]
- Martin Goldstern and Saharon Shelah.
Many simple cardinal invariants. Archive for Mathematical Logic, 32:203-221, 1993. math.LO/9205208. MR 94c:03064
- [J]
- Thomas Jech.
Set theory. Academic Press, New York, 1978. MR 80a:03062
- [KS81]
- Jussi Ketonen and Robert Solovay.
Rapidly growing Ramsey functions. Ann. Math., II. Ser., 113:267-314, 1981. MR 84c:03100
- [KP82]
- Laurie Kirby and Jeff Paris.
Accessible independence results for Peano arithmetic. Bull. Lond. Math. Soc., 14:285-293, 1982. MR 83j:03096
- [Mg2]
- Menachem Magidor.
Chang's conjecture and powers of singular cardinals. J. Symbolic Logic, 42:272-276, 1977. MR 58:5221
- [Mg1]
- Menachem Magidor.
On the singular cardinals problem II. Annals Math., 106:517-547, 1977.
- [FOM]
- Stephen G. Simpson (moderator).
E-mail list for discussing Foundations of Mathematics. http://www.math.psu.edu/simpson/fom/.
- [PH77]
- Jeff Paris and Leo Harrington.
A mathematical incompleteness in Peano Arithmetic. In Jon Barwise, editor, Handbook of Mathematical Logic, pages 1133-1142. North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. MR 56:15351
- [Sh:E12]
- Saharon Shelah.
Analytical Guide and Corrections to []. math.LO/9906022.
- [Sh:E16]
- Saharon Shelah.
The future of set theory. In Haim Judah, editor, Set Theory of the Reals, volume 6 of Israel Mathematical Conference Proceedings. Proceedings of the Winter Institute held at Bar-Ilan University, Ramat Gan, January 1991. MR 94i:03002
- [Sh 413]
- Saharon Shelah.
More Jonsson Algebras. Archive for Mathematical Logic, accepted. math.LO/9809199.
- [Sh 724]
- Saharon Shelah.
On nice equivalence relations on . Archive for Mathematical Logic, accepted. math.LO/0009064.
- [Sh 702]
- Saharon 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]
- Saharon Shelah.
Polish Algebras shy from freedom. Israel Journal of Mathematics, submitted.
- [Sh 705]
- Saharon Shelah.
Toward classification theory of good frames and abstract elementary classes, preprint.
- [Sh:b]
- Saharon Shelah.
Proper forcing, volume 940 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, xxix+496 pp, 1982. MR 84h:03002
- [Sh 170]
- Saharon Shelah.
On logical sentences in PA. In Logic colloquium '82 (Florence, 1982), volume 112 of Stud. Logic Found. Math, pages 145-160. North-Holland, Amsterdam-New York, 1984. MR 86g:03097
- [Sh 111]
- Saharon Shelah.
On power of singular cardinals. Notre Dame Journal of Formal Logic, 27:263-299, 1986. MR 87j:04006
- [Sh 300]
- Saharon Shelah.
Universal classes. In Classification theory (Chicago, IL, 1985), volume 1292 of Lecture Notes in Mathematics, pages 264-418. Springer, Berlin, 1987. Proceedings of the USA-Israel Conference on Classification Theory, Chicago, December 1985; ed. Baldwin, J.T. MR 91k:03088
- [Sh:g]
- Saharon Shelah.
Cardinal Arithmetic, volume 29 of Oxford Logic Guides. Oxford University Press, 1994. MR 96e:03001
- [Sh:f]
- Saharon Shelah.
Proper and improper forcing. Perspectives in Mathematical Logic. Springer, 1998. MR 98m:03002
- [Sh 666]
- Saharon Shelah.
On what I do not understand (and have something to say). Fundamenta Mathematicae, 166:1-82, 2000. math.LO/9906113.
- [Sh 460]
- Saharon Shelah.
The Generalized Continuum Hypothesis revisited. Israel Journal of Mathematics, 116:285-321, 2000. math.LO/9809200. MR 2001g:03095
- [ShSt 419]
- Saharon Shelah and Lee Stanley.
Filters, Cohen Sets and Consistent Extensions of the Erdos-Dushnik-Miller Theorem. Journal of Symbolic Logic, 65:259-271, 2000. math.LO/9709228. MR 2001j:03085
- [Sh:E25]
- Shelah, Saharon.
You Can Enter Cantor's Paradise! In Proceedings of the Erdos Conference. Budapest'99. math.LO/0102056.
- [Si]
- Jack Silver.
On the singular cardinal problem. In Proceedings of the International Congress of Mathematicians, volume I, pages 265-268, Vancouver, 1974. MR 55:2576
- [Wd00]
- Hugh Woodin.
The Continuum Hypothesis, Part I. Notices of the American Mathematical Society, 48:567-576, 2001.
- [Woxx]
- Hugh 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
Email:
shelah@math.huji.ac.il
DOI:
http://dx.doi.org/10.1090/S0273-0979-03-00981-9
PII:
S 0273-0979(03)00981-9
Keywords:
Mathematical logic,
set theory,
independence,
incompleteness,
forcing,
large cardinals
Posted:
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
Article copyright:
© Copyright 2003 American Mathematical Society
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