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On strong chains of uncountable functions

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Abstract

For functionsf,g:ω1 → ω1, where ω1 is the first uncountable cardinal, we write thatf≪g if and only if {ξ ∈ ω1 :f(ξ)≥g(ξ)} is finite. We prove the consistency of the existence of a well-ordered increasing ≪-chain of length ω12, solving a problem of A. Hajnal. The methods previously developed by us involveforcing with side conditions in morasses which is a variation on Todorcevic'sforcing with models as side conditions. The paper is self-contained and requires from the reader knowledge of Kunen's textbook and some basic experience with proper forcing and elementary submodels.

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References

  • [B1] J. Baumgartner,Almost disjoint sets, the dense set problem and the partition calculus, Annals of Mathematical Logic9 (1976), 401–439.

    Article  MATH  MathSciNet  Google Scholar 

  • [B2] J. Baumgartner,Applications of proper forcing, inHandbook of Set-theoretic Topology (K. Kunen and J. Vaughan, eds.), North-Holland, Amsterdam, 1984, pp. 913–959.

    Google Scholar 

  • [BS] J. Baumgartner and S. Shelah,Remarks on superatomic Boolean algebras, Annals of Pure and Applied Logic33 (1987), 109–129.

    Article  MATH  MathSciNet  Google Scholar 

  • [D] K. Devlin,Aspects of Constructibility, Lecture Notes in Mathematics,354, Springer-Verlag, Berlin, 1973.

    MATH  Google Scholar 

  • [DL] H-D. Donder and J-P. Levinski,Some principles related to Chang's Conjecture, Annals of Pure and Applied Logic45 (1989), 39–101.

    Article  MATH  MathSciNet  Google Scholar 

  • [vD] E. van Douwen,Integers and topology, inHandbook of Set-theoretic Topology (K. Kunen and J. Vaughan, eds.), North-Holland, Amsterdam, 1984, pp. 111–167.

    Google Scholar 

  • [J1] R. Jensen,The fine structure of the Constructible Universe, Annals of Pure and Applied Logic4 (1972), 229–308.

    MATH  MathSciNet  Google Scholar 

  • [K] K. Kunen,Set Theory, North-Holland, Amsterdam, 1980.

    MATH  Google Scholar 

  • [Ka] A. Kanamori,Morasses in Combinatorial Set Theory, inSurveys in Set Theory (A. R. D. Mathias, ed.), London Mathematical Society Lecture Note Series 87, Cambridge University Press, 1983, pp. 165–196.

  • [Ko1] P. Koszmider,Semimorasses and nonreflection at singular cardinals, Annals of Pure and Applied Logic72 (1995), 1–23.

    Article  MATH  MathSciNet  Google Scholar 

  • [Ko2] P. Koszmider,On the existence of strong chains in ℘(ω 1 )/Fin, Journal of Symbolic Logic63 (1998), 1055–1062.

    Article  MATH  MathSciNet  Google Scholar 

  • [Ko3] P. Koszmider,A method in generic stepping-up with applications, preprint.

  • [L] R. Laver,Adding dominating functions mod finite, Periodica Mathematica Hungarica35 (1997), 35–41.

    Article  MATH  MathSciNet  Google Scholar 

  • [N] P. Nyikos,Subsets of ω ω and the Frechet-Urysohn and α i , Topology and its Applications48 (1992), 91–116.

    Article  MATH  MathSciNet  Google Scholar 

  • [R] J. Roitman,Height and width of superatomic Boolean algebras, Proceedings of the American Mathematical Society94 (1985), 9–14.

    Article  MATH  MathSciNet  Google Scholar 

  • [Sch] M. Scheepers,Gaps in ω ω, inSet Theory of the Reals (Ramat Gan, 1991), Israel Mathematical Conference Proceedings, Vol. 6, Bar-Ilan University, Ramat Gan, 1993, pp. 439–561.

    Google Scholar 

  • [T1] S. Todorĉević,Directed sets and cofinal types, Transactions of the American Mathematical Society290 (1985), 711–723.

    Article  MathSciNet  Google Scholar 

  • [T2] S. Todorĉević,Partitioning pairs of countable ordinals, Acta Mathematica159 (1987), 261–294.

    Article  MathSciNet  Google Scholar 

  • [T3] S. Todorĉević,Partition problems in topology, Contemporary Mathematics Vol. 84, American Mathematical Society, Providence, RI, 1989.

    Google Scholar 

  • [T4] S. Todorĉević,Remarks on Martin's Axiom and the continuum hypothesis, Canadian Journal of Mathematics43 (1991), 832–851.

    Google Scholar 

  • [V1] D. Velleman,Morasses, diamond and forcing, Annals of Pure and Applied Logic23 (1983), 199–281.

    MathSciNet  Google Scholar 

  • [V2] D. Velleman,Simplified morasses, Journal of Symbolic Logic49 (1984), 257–271.

    Article  MATH  MathSciNet  Google Scholar 

  • [Z] J. Zapletal,Strongly almost disjoint functions, Israel Journal of Mathematics97 (1997), 101–111.

    Article  MATH  MathSciNet  Google Scholar 

  • [Zw] W. Zwicker,℘ κ λ Combinatorics I: Stationary coding sets rationalize the club filter, inAxiomatic Set Theory (J. Baumgartner, D. Martin and S. Shelah, eds.); Contemporary Mathematics Vol. 31, American Mathematical Society, Providence, RI, 1984, pp. 243–259.

    Google Scholar 

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Authors and Affiliations

  1. Departamento de Matemática, Universidade de São Paulo, Caixa Postal: 66281, CEP: 05315-970, São Paulo, SP, Brasil

    Piotr Koszmider

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  1. Piotr Koszmider
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Correspondence to Piotr Koszmider.

Additional information

Some of the research leading to this paper was supported by NSF of USA Grant DMS-9505098 held by the author at Auburn University, AL, USA, some was done while the author was visiting Ohio University at Athens, OH, USA from September 1997 till March 1998, and some was done at Universidade de São Paulo, where the author has been working since 1998. We would like to thank the set-theory groups from these universities for their hospitality.

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Koszmider, P. On strong chains of uncountable functions. Isr. J. Math. 118, 289–315 (2000). https://doi.org/10.1007/BF02803525

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  • DOI: https://doi.org/10.1007/BF02803525

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