Different cofinalities of tree ideals

by Shelah and Spinas. [ShSi:1110]

We introduce a general framework of generalized tree forcings, GTF for short, that includes the classical tree forcings like Sacks, Silver, Laver or Miller forcing. Using this concept we study the cofinality of the ideal {I}(mathbf {Q}) associated with a GTF mathbf {Q} . We show that if for two GTF's mathbf {Q_0} and mathbf {Q_1} the consistency of add({I}(mathbf {Q_0})) < add({I}(mathbf {Q_1})) holds, then we can obtain the consistency of cof({I}(mathbf {Q_1})) < cof({I} (mathbf {Q_0})) . We also show that cof({I}(mathbf {Q})) can consistently be any cardinal of cofinality larger than the continuum.

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