The generalized continuum hypothesis revisited
- Saharon Shelah
- … show all 1 hide
Rent the article at a discountRent now
* Final gross prices may vary according to local VAT.Get Access
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.
- Foreman, M., Woodin, H. (1991) The generalized continuum hypothesis can fail everywhere. Annals of Mathematics 133: pp. 1-36 CrossRef
- Gitik, M., Shelah, S. (1989) On certain, destructibility of strong cardinals and a question of Hajnal. Archive for Mathematical Logic 28: pp. 35-42 CrossRef
- Gregory, J. (1976) Higher Souslin trees and the generalized continuum hypothesis. Journal of Symbolic Logic 41: pp. 663-671 CrossRef
- Hajnal, A., Juhász, I., Shelah, S. (1986) Splitting strongly almost disjoint families. Transactions of the American Mathematical Society 295: pp. 369-387 CrossRef
- Hart, B., Laflamme, C., Shelah, S. (1993) Models with second order properties, V: A General principle. Annals of Pure and Applied Logic 64: pp. 169-194 CrossRef
- Laskowski, M. C., Pillay, A., Rothmaler, P. (1992) Tiny models of categorical theories. Archive for Mathematical Logic 31: pp. 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.
- Shelah, S. (1979) On successors of singular cardinals. Logic Colloquium ’78 (Mons, 1978), Vol. 97 ofStudies in Logic Foundations of Mathematics. North-Holland, Amsterdam-New York, pp. 357-380
- Shelah, S. (1981) Models with second order properties. III. Omitting types for L(Q). Archiv für Mathematische Logik und Grundlagenforschung 21: pp. 1-11 CrossRef
- Shelah, S. (1991) Reflecting stationary sets and successors of singular cardinals. Archive for Mathematical Logic 31: pp. 25-53 CrossRef
- Shelah, S. (1992) Cardinal arithmetic for skeptics. American Mathematical Society. Bulletin. New Series 26: pp. 197-210
- Shelah, S. Advances in cardinal arithmetic. In: Sauer, N.W. eds. (1993) Finite and Infinite Combinatorics in Sets and Logic. Kluwer Academic Publishers, Dordrecht, pp. 355-383
- Shelah, S. (1993) More on cardinal arithmetic. Archive for Mathematical Logic 32: pp. 399-428 CrossRef
- [Sh:g] S. Shelah,Cardinal Arithmetic, Vol. 29 ofOxford Logic Guides, Oxford University Press, 1994.
- Shelah, S. (1994) Cardinalities of topologies with small base. Annals of Pure and Applied Logic 68: pp. 95-113 CrossRef
- Shelah, S. (1996) Further cardinal arithmetic. Israel Journal of Mathematics 95: pp. 61-114 CrossRef
- The generalized continuum hypothesis revisited
Israel Journal of Mathematics
Volume 116, Issue 1 , pp 285-321
- Cover Date
- Print ISSN
- Online ISSN
- Additional Links
- Industry Sectors
- 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