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Further cardinal arithmetic

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Abstract

We continue the investigations in the author’s book on cardinal arithmetic, assuming some knowledge of it. We deal with the cofinality of (S ≤ℵ 0(κ), ⊆) for κ real valued measurable (Section 3), densities of box products (Section 5,3), prove the equality cov(λ,λ,θ+,2) in more cases even when cf(λ)=ℵ0 (Section 1), deal with bounds of pp(λ) for λ limit of inaccessible (Section 4) and give proofs to various claims I was sure I had already written but did not find (Section 6).

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References

  • [ARSh153] U. Abraham, M. Rubin and S. Shelah,On the consistency of some partition theorems for continuous colorings and the structure of ℵ 1-dense real order type, Annals of Pure and Applied Logic29 (1985), 123–206.

    Article  MATH  MathSciNet  Google Scholar 

  • [BoSh210] R. Bonnet and S. Shelah,Narrow Boolean algebras, Annals of Pure and Applied Logic28 (1985), 1–12.

    Article  MATH  MathSciNet  Google Scholar 

  • [CEG] F. S. Cater, Paul Erdös and Fred Galvin,On the density of λ-box products, General Topology and its Applications9 (1978), 307–312.

    Article  MathSciNet  Google Scholar 

  • [CN1] W. W. Comfort and S. Negrepontis,On families of large oscillation, Fundamenta Mathematicae75 (1972), 275–290.

    MATH  MathSciNet  Google Scholar 

  • [CN2] W. W. Comfort and S. Negrepontis,The Theory of Ultrafilters, Grundlehren der Mathematischen Wissenschaften 211, Springer-Verlag, Berlin-Heidelberg-New York, 1974.

    MATH  Google Scholar 

  • [CR] W. W. Comfort and Lewis C. Robertson,Cardinality constraints for pseudocompact and for totally dense subgroups of compact topological groups, Pacific Journal of Mathematics119 (1985), 265–285.

    MATH  MathSciNet  Google Scholar 

  • [DoJe] T. Dodd and R. Jensen,The Covering Lemma for K, Annals of Mathematical Logic22 (1982), 1–30.

    Article  MATH  MathSciNet  Google Scholar 

  • [EK] R. Engelking and M. Karlowicz,Some theorems of set theory and their topological consequences, Fundamenta Math57 (1965), 275–285.

    MATH  MathSciNet  Google Scholar 

  • [H] Edwin Hewitt,A remark on density characters, Bulletin of the American Mathematical Society52 (1946), 641–643.

    MATH  MathSciNet  Google Scholar 

  • [Ma] Edward Marczewski,Séparabilité et multiplication cartésienne des espaces topologiques, Fundamenta Mathematicae34 (1947), 127–143.

    MATH  MathSciNet  Google Scholar 

  • [Mo] D. Monk,Cardinal functions of Boolean algebras, circulated notes.

  • [P] E. S. Pondiczery,Power problems in abstract spaces, Duke Mathematical Journal11 (1944), 835–837.

    Article  MATH  MathSciNet  Google Scholar 

  • [Sh-b] S. Shelah,Proper Forcing, Lectures Notes in Mathematics940, Springer-Verlag, Berlin-New York, 1982.

    MATH  Google Scholar 

  • [Sh-f] S. Shelah,Proper and Improper Forcing, Perspectives in Mathematical Logic, to appear.

  • [Sh-g] S. Shelah,Cardinal Arithmetic, Oxford University Press, 1994.

  • [Sh88] S. Shelah,Classification theory for non-elementary classes II, inAbstract Elementary Classes, Proc. U.S.A-Israel Conference on Classification Theory, Chicago 12/85 (J. T. Baldwin, ed.), Springer-Verlag Lecture Notes1292 (1987), 419–497.

  • [Sh92] S. Shelah,Remarks on Boolean algebras, Algebra Universalis11 (1980), 77–89.

    Article  MATH  MathSciNet  Google Scholar 

  • [Sh93] S. Shelah,Simple unstable theories, Annals of Mathematical Logic19 (1980), 177–203.

    Article  MATH  MathSciNet  Google Scholar 

  • [Sh107] S. Shelah,Models with second order properties IV, a general method and eliminating diamonds, Annals of Mathematical Logic25 (1983), 183–212.

    MATH  Google Scholar 

  • [Sh111] S. Shelah,On powers of singular cardinals, Notre Dame Journal of Formal Logic27 (1986), 263–299.

    Article  MATH  MathSciNet  Google Scholar 

  • [RuSh117] M. Rubin and S. Shelah,Combinatorial problems on trees: partitions, Δ-systems and large free subtrees, Annals of Pure and Applied Logic33 (1987), 43–81.

    Article  MATH  MathSciNet  Google Scholar 

  • [ShLH162] S. Shelah, C. Laflamme and B. Hart,Models with second order properties V: a general principle, Annals of Pure and Applied Logic64 (1993), 169–194.

    Article  MATH  MathSciNet  Google Scholar 

  • [Sh233] S. Shelah,Remarks on the number of ideals of Boolean algebra and open sets of a topology, Springer-Verlag Lecture Notes1182 (1986), 151–187.

    Google Scholar 

  • [Sh282] S. Shelah,Successors of singulars, cofinalities of reduced products of cardinals and productivity of chain conditions, Israel Journal of Mathematics62 (1988), 213–256.

    MATH  MathSciNet  Google Scholar 

  • [Sh288] S. Shelah,Strong partition relations below the power set: Consistency, was Sierpinski right, II?, Proceedings of the Conference on Set Theory and its Applications in honor of A. Hajnal and V. T. Sos, Budapest, 1/91, Colloquia Mathematica Societatis Janos Bolyai, Sets, Graphs and Numbers, Vol. 60, 1991.

  • [Sh300] S. Shelah,Universal classes, Chapter I–IV, Proceedings of the USA-Israel Conference on Classification Theory, Chicago, December 1985 (J. Baldwin, ed.), Springer-Verlag Lecture Notes1292 (1987), 264–418.

    Google Scholar 

  • [Sh326] S. Shelah,Viva la difference I: Nonisomorphism of ultrapowers of countable models, inSet Theory of the Continuum, Mathematical Sciences Research Institute Publications, Vol. 26, Springer-Verlag, Berlin, 1992, pp. 357–405.

    Google Scholar 

  • [GiSh344] M. Gitik and S. Shelah,On certain indestructibility of strong cardinals and a question of Hajnal, Archive for Mathematical Logic28 (1989), 35–42.

    Article  MATH  MathSciNet  Google Scholar 

  • [Sh345] S. Shelah,Products of regular cardinals and cardinal invariants of products of Boolean algebras, Israel Journal of Mathematics70 (1990), 129–187.

    MATH  MathSciNet  Google Scholar 

  • [Sh345a] S. Shelah,Basic: Cofinalities of small reduced products, inCardinal Arithmetic, Chapter I, Oxford University Press, 1994.

  • [Sh345b] S. Shelah,Entangled orders and narrow Boolean algebras, inCardinal Arithmetic, Appendix 2, Oxford University Press, 1994.

  • [Sh355] S. Shelah, ℵω+1 has a Jonsson algebra, inCardinal Arithmetic, Chapter II, Oxford University Press, 1994.

  • [Sh365] S. Shelah,There are Jonsson algebras in many inaccessible cardinals, inCardinal Arithmetic, Chapter III, Oxford University Press, 1994.

  • [Sh371] S. Shelah,Advanced: cofinalities of small reduced products, inCardinal Arithmetic, Chapter VIII, Oxford University Press, 1994.

  • [Sh386] S. Shelah,Bounding pp(μ) when cf(μ) > μ > ℵ 0 using ranks and normal ideals, inCardinal Arithmetic, Chapter V, Oxford University Press, 1994.

  • [Sh400] S. Shelah,Cardinal arithmetic, inCardinal Arithmetic, Chapter IX, Oxford University Press, 1994.

  • [Sh410] S. Shelah,More on cardinal arithmetic, Archive for Mathematical Logic32 (1993), 399–428.

    Article  MATH  MathSciNet  Google Scholar 

  • [GiSh412] M. Gitik and S. Shelah,More on ideals with simple forcing notions, Annals of Pure and Applied Logic59 (1993), 219–238.

    Article  MATH  MathSciNet  Google Scholar 

  • [Sh420] S. Shelah,Advances in cardinal arithmetic, Proceedings of the Banff Conference in Alberta; 4/91,Finite and Infinite Combinatorics in Sets and Logic (N. W. Saure et al., eds.), Kluwer Academic Publishers, Dordrecht 1993, pp. 355–383.

    Google Scholar 

  • [Sh460] S. Shelah,The Generalized Continuum Hypothesis revisited, submitted to Israel Journal of Mathematics.

  • [Sh513] S. Shelah,PCF and infinite free subsets, submitted to Archive for Mathematical Logic.

  • [Sh589] S. Shelah,PCF theory: applications, in preparation.

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Authors and Affiliations

  1. Institute of Mathematics, The Hebrew University of Jerusalem, Jerusalem, Israel

    Saharon Shelah

  2. Department of Mathematics, Rutgers University, New Brunswick, NJ, USA

    Saharon Shelah

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  1. Saharon Shelah
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Additional information

Done mainly 1–4/1991. I thank Alice Leonhardt for typing and retyping so beautifully and accurately. Partially supported by the Basic Research Fund, Israel Academy of Sciences. Publication number 430.

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Shelah, S. Further cardinal arithmetic. Israel J. Math. 95, 61–114 (1996). https://doi.org/10.1007/BF02761035

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  • DOI: https://doi.org/10.1007/BF02761035

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