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Decisive creatures and large continuum

Published online by Cambridge University Press:  12 March 2014

Jakob Kellner and
Saharon Shelah
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, IsraelDepartment of Mathematics, Rutgers University, New Brunswick, Nj 08854, USA, E-mail: shelah@math.huji.ac.il, URL: http://shelah.logic.at
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Abstract

For f, g ∈ ωω 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.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 1 , March 2009, pp. 73 - 104
Copyright
Copyright © Association for Symbolic Logic 2009

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References

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