The short extenders gap two forcing is of Prikry type
Suppose \(\kappa = \bigcup_{n<\omega} \kappa_n\), where \(\langle\kappa_n: n<\omega\rangle\) is an increasing sequence of cardinals with the property that for each \(n<\omega\), there is a \(\langle\kappa_n, \kappa_n^{+n+2}\rangle\) extender. Let \((\mathbb P,\leq,\leq^\ast)\) be the associated short extender gap-2 notion of forcing. This forcing is due to \textit{M. Gitik} [Ann. Pure Appl. Logic 80, No.~1, 17--33 (1996; Zbl 0849.03039)], who proved that \(\mathbb P\) preserves cardinals and adds no new bounded subsets of \(\kappa\), and that in \(V^{\mathbb P}\), \(2^\kappa = \kappa^{++}\). The author establishes that \((\mathbb P,\leq, \leq^\ast)\) is of Prikry type, which means that for each condition \(p\in\mathbb P\) and each formula \(\sigma\) in the \(\mathbb P\)-forcing language, there is \(p^\ast \leq^\ast p\) deciding \(\sigma\).
scientific article; zbMATH DE number 5646249