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The club principle and the distributivity number

  • Heike Mildenberger (a1)


We give an affirmative answer to Brendle's and Hrušák's question of whether the club principle together with is consistent. We work with a class of axiom A forcings with countable conditions such that is determined by finitely many elements in the conditions p and q and that all strengthenings of a condition are subsets, and replace many names by actual sets. There are two types of technique: one for tree-like forcings and one for forcings with creatures that are translated into trees. Both lead to new models of the club principle.



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[1]Bartoszyński, Tomek and Judah, Haim, Set theory, on the structure of the real line, A K Peters, 1995.
[2]Baumgartner, James E. and Laver, Richard, Iterated perfect-set forcing, Annals of Mathematical Logic, vol. 17 (1979), pp. 271288.
[3]Blass, A., Combinatorial cardinal characteristics of the continuum, Handbook of set theory (Foreman, Matthew and Kanamori, Akihiro, editors), vol. 1, Springer, 2010, pp. 395490.
[4]Blass, Andreas, Applications of superperfect forcing and its relatives, Set theory and its applications (Steprāns, Juris and Watson, Steve, editors), Lecture Notes in Mathematics, vol. 1401, 1989, pp. 1840.
[5]Blass, Andreas and Shelah, Saharon, There may be simple P1- and P2-points and the Rudin-Keisler ordering may be downward directed, Annals of Pure and Applied Logic vol. 33 (1987), pp. 213243.
[6]Brendle, Jörg, Cardinal invariants of the continuum and combinatorics on uncountable cardinals, Annals of Pure and Applied Logic, vol. 144 (2006), pp. 4372.
[7]Broverman, Samuel, Ginsburg, John, Kunen, Kenneth, and Tall, Franklin, Topologies determined by σ-ideals on ω1, Canadian Journal of Mathematics, vol. 30 (1978), pp. 13061312.
[8]Devlin, Keith J. and Shelah, Saharon, A weak version of ♢ which follows from 20 < 21, Israel Journal of Mathematics, vol. 29 (1978), pp. 239247.
[9]Džamonja, Mirna and Shelah, Saharon, ♣ does not imply the existence of a Suslin tree, Israel Journal of Mathematics, vol. 113 (1999), pp. 163204.
[10]Fuchino, Sakaé, Mildenberger, Heike, Shelah, Saharon, and Vojtáš, Peter, On absolutely divergent series, Fundamenta Mathematicae, vol. 160 (1999), pp. 255268.
[11]Fuchino, Sakaé, Shelah, Saharon, and Soukup, Lajos, Sticks and clubs, Annals of Pure and Applied Logic, vol. 90 (1997), pp. 5777, math.LO/9804153.
[12]Hrušák, Michal, Life in the Sacks model, Acta Universitatis Carolinae. Mathematica et Physica, vol. 42 (2001), pp. 4358.
[13]Jech, Thomas, Set theory, Springer, 2003, the Third Millenium Edition, revised and expanded.
[14]Moore, Justin Tatch, Hrušák, Michael, and Džamonja, Mirna, Parametrized ♢ principles, Transactions of the American Mathematical Society, vol. 356 (2004), pp. 22812306.
[15]Ostaszewski, A. J., A perfectly normal countably compact scattered space which is not strongly zero-dimensional, Journal of the London Mathematical Society. Second Series, vol. 14 (1976), no. 1, pp. 167177.
[16]Rosłanowski, Andrzej and Shelah, Saharon, Norms on possibilities. I. Forcing with trees and creatures, Memoirs of the American Mathematical Society, vol. 141 (AMS, 1999), no. 671.
[17]Shelah, Saharon, Proper and improper forcing, 2nd ed., Springer, 1998.
[18]Truss, J. K., The noncommutativity of random and generic extensions, this Journal, vol. 48 (1983), no. 4, pp. 10081012.


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The club principle and the distributivity number

  • Heike Mildenberger (a1)


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