Killing GCH everywhere by a cofinality-preserving forcing notion over a model of GCH (Q2866774)

The investigation of the global behavior of the power function is a long standing project in set theory. A seminal precursor of this paper is [\textit{C. Merimovich}, J. Symb. Log. 72, No. 2, 361--417 (2007; Zbl 1153.03036)], in which, using the extender-based Radin forcing, Merimovich built a model in which at every \(\lambda\), \(2^\lambda=\lambda^{+n}\) for some fixed finite \(n>1\). NEWLINENEWLINENEWLINEThe main result in this paper is:NEWLINENEWLINE{ Theorem 1.1}. Assume \(\text{GCH}\) and the existence of a \((\kappa+4)\)-strong cardinal \(\kappa\). Then, there is a pair \(V_1 \subseteq V_2\) of \text{ZFC} models with the same cardinals and cofinalities, and such that \(V_1 \models {}\)\text{GCH} and \(V_2 \models \forall \lambda\,(2^\lambda = \lambda^{+3})\).NEWLINENEWLINEThis result improves that of Merimovich in that the extension is cofinality-preserving. This is achieved by a variation of Merimovich's construction: at the preparation stage, a model \(V\) is obtained which only forces \(2^\alpha = \alpha^{+3}\) at the first three successors of \(\kappa\) without adding new subsets of \(\kappa\). This is for preserving \text{GCH} below \(\kappa\). The second step gives \(V_2\), which is essentially the same as Merimovich's extender-based Radin forcing with interleaved Levy collapses, but extra work is needed for getting the generic that guides the interleaved collapses. \(V_1\) is the intermediate model obtained by the ordinary Radin forcing with interleaved collapses (guided by the collapsing part of the guiding generic). The key point is that Merimovich's extender-based Radin forcing with collapses can be projected into the ordinary Radin forcing with collapses. This ensures that \(V_1 \models {}\)\text{GCH} below \(\kappa\) while having the same cofinalities and cardinalities as in \(V_2\).NEWLINENEWLINEThe authors remark that the hypothesis is slightly overstated, the exact strength for a fixed \(3\)-gap is actually a cardinal \(\kappa\) with \(o(\kappa)=\kappa^{+3}+\kappa^+\), and the argument can extend the result to all finite \(n\geq 3\).NEWLINENEWLINEThe paper is highly technical, although necessary definitions are included, advanced knowledge of Prikry-type forcing, in particular, extender-based Radin forcing, is necessary.