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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Lipparini, Paolo 2010. More on regular and decomposable ultrafilters in ZFC. Mathematical Logic Quarterly, Vol. 56, Issue. 4, p. 340.


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    Asperó, David 2009. On a convenient property about $${[\gamma]^{\aleph_0}}$$. Archive for Mathematical Logic, Vol. 48, Issue. 7, p. 653.


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    Larson, Paul B. 2005. The canonical function game. Archive for Mathematical Logic, Vol. 44, Issue. 7, p. 817.


    LARSON, PAUL and SHELAH, SAHARON 2003. BOUNDING BY CANONICAL FUNCTIONS, WITH CH. Journal of Mathematical Logic, Vol. 03, Issue. 02, p. 193.


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Canonical functions, non-regular ultrafilters and Ulam's problem on ω1

  • Oliver Deiser (a1) and Dieter Donder (a2)
  • DOI: http://dx.doi.org/10.2178/jsl/1058448434
  • Published online: 12 March 2014
Abstract
Abstract

Our main results are:

Theorem 1. Con(ZFC + “every function f: ω1ω1 is dominated by a canonical function”) implies Con(ZFC + “there exists an inaccessible limit of measurable cardinals”). [In fact equiconsistency holds.]

Theorem 3. Con(ZFC + “there exists a non-regular uniform ultrafilter on ω1”) implies Con(ZFC + “there exists an inaccessible stationary limit of measurable cardinals”).

Theorem 5. Con (ZFC + “there exists an ω1-sequence of ω1-complete uniform filters on ω1 s.t. every A ⊆ ω1 is measurable w.r.t. a filter in (Ulam property)”) implies Con(ZFC + “there exists an inaccessible stationary limit of measurable cardinals”).

We start with a discussion of the canonical functions and look at some combinatorial principles. Assuming the domination property of Theorem 1, we use the Ketonen diagram to show that ω2V is a limit of measurable cardinals in Jensen's core model KMO for measures of order zero. Using related arguments we show that ω2V is a stationary limit of measurable cardinals in KMO, if there exists a weakly normal ultrafilter on ω1. The proof yields some other results, e.g., on the consistency strength of weak*-saturated filters on ω1, which are of interest in view of the classical Ulam problem.

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[4]H.-D. Donder , Families of almost disjoint functions, Contemporary Mathematics, vol. 31 (1984), pp. 7178.

[7]H.-D. Donder and P. Koepke , On the consistency strength of ‘accessible’ Jonsson cardinals and of the weak Chang conjecture, Annals of Pure and Applied Logic, vol. 25 (1983), pp. 233261.

[9]M. Gitik and S. Shelah , Forcing with ideals and simple forcing notions, Israel Journal of Mathematics, vol. 68 (1989), pp. 129160.

[12]A. Kanamori , Weakly normal filters and irregular ultrafilters, Transactions of the American Mathematical Society, vol. 220 (1976), pp. 393399.

[14]J. Ketonen , Non-regular ultrafilters and large cardinals. Transactions of the American Mathematical Society, vol. 224 (1976), pp. 6173.

[16]S. Shelah , Proper and improper forcing, 2nd ed., Perspectives in Mathematical Logic, Springer, Berlin, 1998.

[17]A.D. Taylor , Regularity properties of ideals and ultrafilters. Annals of Mathematical Logic, vol. 16 (1979), pp. 3355.

[20]W.H. Woodin , The axiom of determinacy, forcing axioms, and the nonstationary ideal, de Gruyter Series in Logic and Its Applications, vol. 1, Walter de Gruyter, Berlin, 1999.

[21]M. Zeman , Inner models and large cardinals, de Gruyter Series in Logic and Its Applications, vol. 5, Walter de Gruyter, Berlin, 2002.

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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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