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Cardinal arithmetic for skeptics
Author:
Saharon Shelah
Journal:
Bull. Amer. Math. Soc. 26 (1992), 197-210
MSC (2000):
Primary 03E05; Secondary 03E10, 03E35, 04A10, 06E05, 08A05, 20A15, 54A25
MathSciNet review:
1112424
Full-text PDF
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Similar Articles |
Additional Information
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Uri
Abraham, Matatyahu
Rubin, and Saharon
Shelah, On the consistency of some partition theorems for
continuous colorings, and the structure of ℵ₁-dense real
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partition calculus, Ann. Math. Logic 9 (1976),
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Prikry, On a theorem of Silver, Discrete Math.
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Robert
Bonnet and Saharon
Shelah, Narrow Boolean algebras, Ann. Pure Appl. Logic
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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
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Dodd and R.
Jensen, The core model, Ann. Math. Logic 20
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Matthew
Foreman and W.
Hugh Woodin, The generalized continuum hypothesis can fail
everywhere, Ann. of Math. (2) 133 (1991), no. 1,
1–35. MR
1087344 (91k:03130), http://dx.doi.org/10.2307/2944324
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Galvin and András
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Moti
Gitik, The negation of the singular cardinal hypothesis from
𝑜(𝜅)=𝜅⁺⁺, Ann. Pure Appl. Logic
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-, The strength of the failure of the singular cardinal hypothesis, preprint.
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M. Gitik and M. Magidor, The singular cardinal hypothesis revisited (in preparation).
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Moti
Gitik and Saharon
Shelah, On certain indestructibility of strong cardinals and a
question of Hajnal, Arch. Math. Logic 28 (1989),
no. 1, 35–42. MR 987765
(90e:03063), http://dx.doi.org/10.1007/BF01624081
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M.
Gitik and S.
Shelah, More on simple forcing notions and forcings with
ideals, Ann. Pure Appl. Logic 59 (1993), no. 3,
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Thomas
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Prikry, Ideals over uncountable sets: application of almost
disjoint functions and generic ultrapowers, Mem. Amer. Math. Soc.
18 (1979), no. 214, iii+71. MR 519927
(80f:03059)
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Thomas
Jech and Saharon
Shelah, On a conjecture of Tarski on products
of cardinals, Proc. Amer. Math. Soc.
112 (1991), no. 4,
1117–1124. MR 1070525
(91j:03066), http://dx.doi.org/10.1090/S0002-9939-1991-1070525-1
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Menachem
Magidor, On the singular cardinals problem. I, Israel J. Math.
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Menachem
Magidor, On the singular cardinals problem. II, Ann. of Math.
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(58 #10449b)
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Menachem
Magidor, Chang’s conjecture and powers of singular
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272–276. MR 0485375
(58 #5221)
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William
J. Mitchell, The core model for sequences of measures. I,
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735366 (85i:03163), http://dx.doi.org/10.1017/S030500410006151X
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K. Prikry, Changing measurable to accessible cardinals, Rozprawy Mat. LXVIII (1970), 1-52.
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S.
Shelah, Jonsson algebras in successor cardinals, Israel J.
Math. 30 (1978), no. 1-2, 57–64. MR 0505434
(58 #21572)
- [Sh 71]
Saharon
Shelah, A note on cardinal exponentiation, J. Symbolic Logic
45 (1980), no. 1, 56–66. MR 560225
(82c:03070), http://dx.doi.org/10.2307/2273354
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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 b]
Saharon
Shelah, Proper forcing, Lecture Notes in Mathematics,
vol. 940, Springer-Verlag, Berlin, 1982. MR 675955
(84h:03002)
- [Sh 137]
Saharon
Shelah, The singular cardinals problem: independence results,
Surveys in set theory, London Math. Soc. Lecture Note Ser., vol. 87,
Cambridge Univ. Press, Cambridge, 1983, pp. 116–134. MR 823777
(87b:03114), http://dx.doi.org/10.1017/CBO9780511758867.004
- [Sh 161]
Saharon
Shelah, Incompactness in regular cardinals, Notre Dame J.
Formal Logic 26 (1985), no. 3, 195–228. MR 796637
(87f:03095), http://dx.doi.org/10.1305/ndjfl/1093870869
- [Sh 256]
Saharon
Shelah, More on powers of singular cardinals, Israel J. Math.
59 (1987), no. 3, 299–326. MR 920498
(89h:03082), http://dx.doi.org/10.1007/BF02774143
- [Sh 280]
Saharon
Shelah, Strong negative partition above the continuum, J.
Symbolic Logic 55 (1990), no. 1, 21–31. MR 1043541
(91b:03083), http://dx.doi.org/10.2307/2274951
- [Sh 282]
-, Successors of singulars, productivity of chain conditions, and cofinalities of reduced products of cardinals, Israel J. Math. 60 (1987), 146-166.
- [Sh 327]
-, Strong negative partition relations below the continuum, Acta Math. Hungar. in press.
- [Sh 410]
Saharon
Shelah, More on cardinal arithmetic, Arch. Math. Logic
32 (1993), no. 6, 399–428. MR 1245523
(94j:03100), http://dx.doi.org/10.1007/BF01270465
- [Sh 413]
-, More Jonsson algebras and coloring, preprint.
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-, Advances in cardinal arithmetic, to appear.
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-, Further cardinal arithmetic, preprint.
- [Sh-g]
-, Cardinal arithmetic, OUP, (to appear).
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Saharon
Shelah and Hugh
Woodin, Forcing the failure of CH by adding a real, J.
Symbolic Logic 49 (1984), no. 4, 1185–1189. MR 771786
(86h:03087), http://dx.doi.org/10.2307/2274270
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Dana
Scott, Measurable cardinals and constructible sets, Bull.
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(1961), 521–524. MR 0143710
(26 #1263)
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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)
- [So]
Robert
M. Solovay, Strongly compact cardinals and the GCH,
Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV,
Univ. California, Berkeley, Calif., 1971), Amer. Math. Soc., Providence,
R.I., 1974, pp. 365–372. MR 0379200
(52 #106)
- [T]
Stevo
Todorčević, Remarks on chain conditions in
products, Compositio Math. 55 (1985), no. 3,
295–302. MR
799818 (87b:04003)
- [T1]
Stevo
Todorčević, Remarks on cellularity in products,
Compositio Math. 57 (1986), no. 3, 357–372. MR 829326
(88h:54009)
- [W]
H. Woodin, The Collected Unwritten Works of H. Woodin.
- [ARSh 153]
- U. Abraham, M. Rubin, and S. Shelah, On the consistency of some partition theorems for continuous colorings and the structure of
-dense real order types, Ann. Pure Appl. Logic 29 (1985), 123-206. MR 801036 (87d:03132)
- [B]
- J. Baumgartner, Almost disjoint sets, the dense set problem, and the partition calculus, Ann. Pure Appl. Logic 9 (1975), 401-439. MR 0401472 (53:5299)
- [BP]
- J. Baumgartner and K. Prikry, On a theorem of Silver, Discrete Math. 14 (1976), 17-22. MR 0387061 (52:7908)
- [BoSh 210]
- R. Bonnet and S. Shelah, Narrow boolean algebras, Ann. Pure Appl. Logic 28 (1985), 1-12. MR 776283 (86g:06024)
- [C]
- P. Cohen, Set theory and the continuum hypothesis, Benjamin, NY, 1966. MR 0232676 (38:999)
- [Cu]
- J. Cummings, Consistency results in cardinal arithmetic, Ph. D. thesis, Cal-Tech, Pasadena, CA.
- [CW]
- J. Cummings and H. Woodin, preprint.
- [DeJ]
- K. Devlin and R. B. Jensen, Marginalia to a theorem of Silver, Proceedings of the Logic Colloquium, Kiel 1974 (G. H. Müller, A. Oberschelp, and K. Potthoff, eds.), Lecture Notes in Math., vol. 499, Springer-Verlag, Berlin, 1975, pp. 115-142. MR 0480036 (58:235)
- [DoJ]
- A. Dodd and R. B. Jensen, The core model, Annals Math. Logic (Ann. Pure Appl. Logic) 20 (1981), 43-75. MR 611394 (82i:03063)
- [FW]
- M. Foreman and H. Woodin, G.C.H can fail everywhere, Ann. of Math. (2) 133 (1991), 1-35. MR 1087344 (91k:03130)
- [GH]
- F. Galvin and A. Hajnal, Inequalities for cardinal powers, Ann. of Math. (2) 101 (1975), 491-498. MR 0376359 (51:12535)
- [GHS]
- J. Gertlis, A. Hajnal, and Z. Szentmiklossy, On the cardinality of certain Hausdorff spaces, Frolic Memorial Volume, accepted.
- [Gi]
- M. Gitik, The negation of SCH from
, Ann. Pure Appl. Logic 43 (1989), 209-234. MR 1007865 (90h:03037)
- [Gi1]
- -, The strength of the failure of the singular cardinal hypothesis, preprint.
- [GM]
- M. Gitik and M. Magidor, The singular cardinal hypothesis revisited (in preparation).
- [GiSh 344]
- M. Gitik and S. Shelah, On certain indestructibility of stray cardinals and a question of Hajnal, Arch. Math Logic 28 (1989), 35-42. MR 987765 (90e:03063)
- [GiSh 412]
- M. Gitik and S. Shelah, More on ideals with simple forcing notions, Ann. Pure Appl. Logic (to appear). MR 1213273 (94c:03070)
- [JP]
- T. Jech and K. Prikry, Ideals over uncountable sets: applications of almost disjoint sets and generic ultrapowers, Mem. Amer. Math. Soc. 214 (1979). MR 519927 (80f:03059)
- [JeSh 385]
- T. Jech and S. Shelah, On a conjecture of Tarski on products of cardinals, Proc. Amer. Math. Soc. 112 (1991), 1117-1124. MR 1070525 (91j:03066)
- [Mg]
- M. Magidor, On the singular cardinals problem I, Israel J. Math. 28 (1977), 1-31. MR 0491183 (58:10449a)
- [Mg1]
- -, On the singular cardinals problem II, Ann. of Math. (2) 106 (1977), 517-547. MR 0491184 (58:10449b)
- [Mg2]
- -, Chang's conjecture and powers of singular cardinals, J. Symbolic Logic 42 (1977), 272-276. MR 0485375 (58:5221)
- [M]
- W. Mitchell, The core model for sequences of measures I, Math. Proc. Cambridge Philos. Soc. 95 (1984), 229-260. MR 735366 (85i:03163)
- [Pr]
- K. Prikry, Changing measurable to accessible cardinals, Rozprawy Mat. LXVIII (1970), 1-52.
- [Sh 68]
- S. Shelah, Joñsson algebras in successor cardinals, Israel J. Math. 39 (1978), 475-480. MR 0505434 (58:21572)
- [Sh 71]
- -, A note on cardinal exponentiation, J. Symbolic Logic 45 (1980), 56-66. MR 560225 (82c:03070)
- [Sh 111]
- -, On powers of singular cardinals, Notre Dame J. Formal Logic 27 (1986), 263-299. MR 842153 (87j:04006)
- [Sh b]
- -, Proper Forcing, Lecture Notes in Math., vol. 940, Springer-Verlag, Berlin, 1982. MR 675955 (84h:03002)
- [Sh 137]
- -, The singular cardinal problem: independence results, Proceedings of a Symposium on Set Theory, Cambridge 1978 (A. Mathias, ed.), London Math Soc. Lecture Notes Ser., vol. 87, Cambridge Univ. Press, Cambridge and New York, 1983, pp. 116-134. MR 823777 (87b:03114)
- [Sh 161]
- -, Incompleteness in regular cardinals, Notre Dame J. Formal Logic 26 (1985), 195-228. MR 796637 (87f:03095)
- [Sh 256]
- -, More on powers of singular cardinals, Israel J. Math. 59 (1987), 263-299. MR 920498 (89h:03082)
- [Sh 280]
- -, Strong negative partition relations above the continuum, J. Symbolic Logic 55 (1990), 21-31. MR 1043541 (91b:03083)
- [Sh 282]
- -, Successors of singulars, productivity of chain conditions, and cofinalities of reduced products of cardinals, Israel J. Math. 60 (1987), 146-166.
- [Sh 327]
- -, Strong negative partition relations below the continuum, Acta Math. Hungar. in press.
- [Sh 410]
- -, More on cardinal arithmetic, Arch. Math. Logic (to appear). MR 1245523 (94j:03100)
- [Sh 413]
- -, More Jonsson algebras and coloring, preprint.
- [Sh 420]
- -, Advances in cardinal arithmetic, to appear.
- [Sh 430]
- -, Further cardinal arithmetic, preprint.
- [Sh-g]
- -, Cardinal arithmetic, OUP, (to appear).
- [ShW 159]
- S. Shelah and H. Woodin, Forcing the failure of the CH, J. Symbolic Logic 49 (1984), 1185-1189. MR 771786 (86h:03087)
- [Sc]
- D. Scott, Measurable cardinals and constructible sets, Bull. Acad. Pol. Sci. Ser. Math. Astron. Phys. 9 (1961), 521-524. MR 0143710 (26:1263)
- [Si]
- J. Silver, On the singular cardinal problem, Proceedings ICM, Vancouver 1974, vol. I, pp. 265-268, MR 0429564 (55:2576)
- [So]
- R. Solovay, Strongly compact cardinals and the GCH, Proceedings of the Tarski Symposium, Berkeley 1971, Proc. Sympos. Pure Math., vol. xxi, Amer. Math. Soc., Providence, RI, 1974, pp. 365-372. MR 0379200 (52:106)
- [T]
- S. Todorčevic, Remarks on chain conditions in products, Compositio Math. 56 (1985), 295-302. MR 799818 (87b:04003)
- [T1]
- -, Remarks on cellularity in products, Compositio Math. 57 (1986), 357-372. MR 829326 (88h:54009)
- [W]
- H. Woodin, The Collected Unwritten Works of H. Woodin.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0273-0979-1992-00261-6
PII:
S 0273-0979(1992)00261-6
Article copyright:
© Copyright 1992 American Mathematical Society
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