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Diagonal Prikry extensions. (English) Zbl 1245.03078

The authors prove some results in singular cardinal combinatorics. Most of the results are connected with Shelah’s pcf theory. This is done by an analysis of the generic extensions of Prikry forcing.
They show that consistently there can be a set which carries two scales, one of which is very good and the other is very far from being very good. They consider the axiom “there is \(j : V \rightarrow M\) with \(V_{\lambda} \subseteq M\)” and show that this axiom is consistent with \(2^{\lambda} > \lambda^+\). This is done using an extender-based Prikry forcing. Further on they show that extender-based Prikry forcing adds a good scale in a canonical way.

MSC:

03E45 Inner models, including constructibility, ordinal definability, and core models
03E04 Ordered sets and their cofinalities; pcf theory
03E35 Consistency and independence results
03E55 Large cardinals
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