Journal of Symbolic Logic

Decisive creatures and large continuum

Jakob Kellner and Saharon Shelah

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Abstract

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

It is consistent that cfε,gε=cfε,gεε for ℵ1 many pairwise different cardinals κε and suitable pairs (fε,gε).

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.

Article information

Source
J. Symbolic Logic, Volume 74, Issue 1 (2009), 73-104.

Dates
First available in Project Euclid: 4 January 2009

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1231082303

Digital Object Identifier
doi:10.2178/jsl/1231082303

Mathematical Reviews number (MathSciNet)
MR2499421

Zentralblatt MATH identifier
1183.03035

Subjects
Primary: 03E17: Cardinal characteristics of the continuum 03E40: Other aspects of forcing and Boolean-valued models

Citation

Kellner, Jakob; Shelah, Saharon. Decisive creatures and large continuum. J. Symbolic Logic 74 (2009), no. 1, 73--104. doi:10.2178/jsl/1231082303. https://projecteuclid.org/euclid.jsl/1231082303


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