Creature forcing and large continuum: the joy of halving
\textit{M. Goldstern} and the second author [Arch. Math. Logic 32, No. 3, 203--221 (1993; Zbl 0786.03030)] introduced cardinal characteristics of the form \(c^\forall(f,g)\) and \(c^\exists(f,g)\) defined as follows. Let \(f,g\in\omega^{\omega}\), an \(\left( f,g\right) \)-slalom is a sequence \(S=\left\langle S\left( n\right) :n\in\omega\right\rangle \) such that \(S\left( n\right) \subseteq f\left( n\right) \) and \(\left| S\left( n\right) \right| \leq g\left( n\right) \). A family \(\mathcal{S}\) of \(\left( f,g\right) \)-slaloms is a \(\left( \forall,f,g\right) \)-cover, if for all \(r\in\prod_{n\in\omega}f\left( n\right) \) there is \(S\in\mathcal{S}\) such that \(r\left( n\right) \in S\left( n\right) \) for all \(n\in\omega\). Also, a family \(\mathcal{S}\) of \(\left( f,g\right) \)-slaloms is a \(\left( \exists,f,g\right) \)-cover, if for all \(r\in\prod_{n\in\omega}f\left( n\right) \) there is \(S\in\mathcal{S}\) such that \(r\left( n\right) \in S\left( n\right) \) for infinitely many \(n\in\omega\). The minimal sizes of such covers are denoted by \(c_{f,g}^{\forall}\) and \(c_{f,g}^{\exists}\), respectively. In the paper under review the authors continue their work initiated in [J. Symb. Log. 74, No. 1, 73--104 (2009; Zbl 1183.03035)]. Here they introduce a new mixed-limit creature forcing construction to show that it is consistent that \(c^\forall(f,g)=c^\exists(f,g)=\kappa_\epsilon\) for continuum many pairwise different cardinals \(\kappa_\epsilon\) and suitable pairs \((f_\epsilon,g_\epsilon)\).
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