Decisive creatures and large continuum

by Kellner and Shelah. [KrSh:872]
J Symbolic Logic, 2009
For f>g in omega^omega let c^{for all}_{f,g} be the minimal number of uniform trees with g-splitting needed to for all^infty-cover the uniform tree with f-splitting. c^{exists}_{f,g} is the dual notion for the exists^infty-cover. Assuming CH and given aleph_1 many (sufficiently different) pairs (f_epsilon,g_epsilon) and cardinals kappa_epsilon such that kappa_epsilon^{aleph_0}= kappa_epsilon, we construct a partial order forcing that c^{exists}_{f_epsilon,g_epsilon}= c^{for all}_{f_epsilon,g_epsilon}= kappa_epsilon . For this, we introduce a countable support semiproduct of decisive creatures with bigness and halving. This semiproduct satisfies fusion, pure decision and continuous reading of names.

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