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Decisive creatures and large continuum

Published online by Cambridge University Press:  12 March 2014

Jakob Kellner
Affiliation:
Kurt Gödel Research Center for Mathematical Logic, Universität Wien, Währinger Straße 25, 1090 Wien, Austria, E-mail: kellner@fsmat.at, URL: http://www.logic.univie.ac.at/~kellner
Saharon Shelah
Affiliation:
Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem, 91904, Israel Department of Mathematics, Rutgers University, New Brunswick, Nj 08854, USA, E-mail: shelah@math.huji.ac.il, URL: http://shelah.logic.at
Corresponding

Abstract

For f, gωω let be the minimal number of uniform g-splitting trees (or: Slaloms) to cover the uniform f-splitting tree, i.e., for every branch v of the f-tree, one of the g-trees contains v. is the dual notion: For every branch v, one of the g-trees guesses v(m) infinitely often.

It is consistent that for ℵ1 many pairwise different cardinals κ and suitable pairs (f, g).

For the proof we use creatures with sufficient bigness and halving. We show that the lim-inf creature forcing satisfies fusion and pure decision. We introduce decisiveness and use it to construct a variant of the countable support iteration of such forcings, which still satisfies fusion and pure decision.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2009

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References

[1]Baumgartner, James E., Iterated forcing, Surveys in set theory, London Mathematical Society Lecture Note Series, vol. 87, Cambridge University Press, Cambridge, 1983, pp. 159.Google Scholar
[2]Goldstern, Martin, Tools for your forcing construction, Set theory of the reals (Ramat Gan, 1991), Israel Mathematical Conference Proceedings, vol. 6, Bar-Ilan University, Ramat Gan, 1993, available at http://info.tuwien.ac.at/goldstern/, pp. 305360.Google Scholar
[3]Goldstern, Martin and Shelah, Saharon, Many simple cardinal invariants, Archive for Mathematical Logic, vol. 32 (1993), no. 3, pp. 203221.CrossRefGoogle Scholar
[4]Kellner, Jakob, Even more simple cardinal invariants, Archive for Mathematical Logic, vol. 47 (2008), no. 5, pp. 503515.CrossRefGoogle Scholar
[5]Rosłanowski, Andrzej and Shelah, Saharon, Norms on possibilities. I. Forcing with trees and creatures, Memoirs of the American Mathematical Society, vol. 141 (1999), no. 671, pp. xii + 167.CrossRefGoogle Scholar
[6]Shelah, Saharon, Proper and improper forcing, second ed., Perspectives in Mathematical Logic, Springer-Verlag, Berlin, 1998.CrossRefGoogle Scholar

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