The generalized continuum hypothesis revisited
- Saharon Shelah
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We can reformulate the generalized continuum problem as: for regular κ<λ we have λ to the power κ is λ, We argue that the reasonable reformulation of the generalized continuum hypothesis, considering the known independence results, is “for most pairs κ<λ of regular cardinals, λ to the revised power of κ is equal to λ”. What is the revised power? λ to the revised power of κ is the minimal cardinality of a family of subsets of λ each of cardinality κ such that any other subset of λ of cardinality κ is included in the union of strictly less than κ members of the family. We still have to say what “for most” means. The interpretation we choose is: for every λ, for every large enoughK < ℶ w . Under this reinterpretation, we prove the Generalized Continuum Hypothesis.
- [AAC90] D. L. Alben, G. L. Alexanderson and C. Reid (eds.),More Mathematical People, Harcourt Brace Jovanovich, 1990.
- [Br] F. E. Browder (ed.),Mathematical developments arising from Hilbert’s Problems, Proceedings of Symposia in Pure Mathematics28 (1974), 421.
- [FW] M. Foreman and H. Woodin,The generalized continuum hypothesis can fail everywhere, Annals of Mathematics133 (1991), 1–36. CrossRef
- [GiSh 344] M. Gitik and S. Shelah,On certain, destructibility of strong cardinals and a question of Hajnal, Archive for Mathematical Logic28 (1989), 35–42. CrossRef
- [Gre] J. Gregory,Higher Souslin trees and the generalized continuum hypothesis, Journal of Symbolic Logic41 (1976), 663–671. CrossRef
- [HJSh 249] A. Hajnal, I. Juhász, and S. ShelahSplitting strongly almost disjoint families, Transactions of the American Mathematical Society295 (1986), 369–387. CrossRef
- [HLSh 162] B. Hart, C. Laflamme, and S. Shelah,Models with second order properties, V: A General principle, Annals of Pure and Applied Logic64 (1993), 169–194. CrossRef
- [LaPiRo] M. C. Laskowski, A. Pillay, and P. Rothmaler,Tiny models of categorical theories, Archive for Mathematical Logic31 (1992), 385–396. CrossRef
- [Sh:E12] S. Shelah,Analytical Guide and Corrections to [Sh:g],
- [Sh 575] S. Shelah,Cellularity of free products of Boolean algebras (or topologies), Fundamenta Mathematica, to appear.
- [Sh 668] S. Shelah,On Arhangelskii’s Problem, e-preprint.
- [Sh 666] S. Shelah,On what I do not understand (and have something to say) I, Fundamenta Mathematica, to appear.
- [Sh 513] S. Shelah,PCF and infinite free subsets, Archive for Mathematical Logic, to appear.
- [Sh 589] S. Shelah,PCF theory: applications, Journal of Symbolic Logic, to appear.
- [Sh 108] S. Shelah,On successors of singular cardinals, inLogic Colloquium ’78 (Mons, 1978), Vol. 97 ofStudies in Logic Foundations of Mathematics, North-Holland, Amsterdam-New York, 1979, pp. 357–380.
- [Sh 82] S. Shelah,Models with second order properties. III. Omitting types for L(Q), Archiv für Mathematische Logik und Grundlagenforschung21 (1981), 1–11. CrossRef
- [Sh 351] S. Shelah,Reflecting stationary sets and successors of singular cardinals, Archive for Mathematical Logic31 (1991), 25–53. CrossRef
- [Sh 400a] S. Shelah,Cardinal arithmetic for skeptics, American Mathematical Society. Bulletin. New Series26 (1992), 197–210.
- [Sh 420] S. Shelah,Advances in cardinal arithmetic, inFinite and Infinite Combinatorics in Sets and Logic (N.W. Sauer et al., eds.), Kluwer Academic Publishers, Dordrecht, 1993, pp. 355–383.
- [Sh 410] S. Shelah,More on cardinal arithmetic, Archive for Mathematical Logic32 (1993), 399–428. CrossRef
- [Sh:g] S. Shelah,Cardinal Arithmetic, Vol. 29 ofOxford Logic Guides, Oxford University Press, 1994.
- [Sh 454a] S. Shelah,Cardinalities of topologies with small base, Annals of Pure and Applied Logic68 (1994), 95–113. CrossRef
- [Sh 430] S. Shelah,Further cardinal arithmetic, Israel Journal of Mathematics95 (1996), 61–114. CrossRef
- The generalized continuum hypothesis revisited
Israel Journal of Mathematics
Volume 116, Issue 1 , pp 285-321
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- Saharon Shelah (1) (2)
- Author Affiliations
- 1. Institute of Mathematics, The Hebrew University of Jerusalem, 91904, Jerusalem, Israel
- 2. Department of Mathematics, Rutgers University, 08854, New Brunswick, NJ, USA