Abstract
For \({f,g\in\omega^\omega}\) let \({c^\forall_{f,g}}\) be the minimal number of uniform g-splitting trees needed to cover the uniform f-splitting tree, i.e., for every branch ν of the f-tree, one of the g-trees contains ν. Let \({c^\exists_{f,g}}\) be the dual notion: For every branch ν, one of the g-trees guesses ν(m) infinitely often. We show that it is consistent that \({c^\exists_{f_\epsilon,g_\epsilon}{=}c^\forall_{f_\epsilon,g_\epsilon}{=}\kappa_\epsilon}\) for continuum many pairwise different cardinals \({\kappa_\epsilon}\) and suitable pairs \({(f_\epsilon,g_\epsilon)}\) . For the proof we introduce a new mixed-limit creature forcing construction.
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Kellner, J., Shelah, S. Creature forcing and large continuum: the joy of halving. Arch. Math. Logic 51, 49–70 (2012). https://doi.org/10.1007/s00153-011-0253-8
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DOI: https://doi.org/10.1007/s00153-011-0253-8