A model in which the base-matrix tree cannot have cofinal branches. (English) Zbl 0637.03049

Dordal, Peter Lars

J. Symb. Log. 52, 651-664 (1987).

Let \([\omega]^{\omega}\) be the set of all infinite subsets of \(\omega\). \(x\subseteq_*y\) means that x-y is finite. The distributivity cardinal \({\mathfrak h}\) is the least \(\kappa\) such that forcing with \(<[\omega]^{\omega},\subseteq_*>\) adds a new function \(f: \kappa\to V\). An \(\alpha\)-tower in \([\omega]^{\omega}\) is a descending \(\alpha\)- sequence \(<x_{\beta}: \beta <\alpha >\) such that there is no \(x\in [\omega]^{\omega}\) almost contained in all the \(x_{\beta}.\)
The author constructs a model of ZFC in which the distributivity cardinal \({\mathfrak h}\) is \(2^{\aleph_ 0}=\aleph_ 2\) and in which there are no \(\aleph_ 2\)-towers in \([\omega]^{\omega}\). This is done by iterating Mathias forcing. Mathias forcing is factorized into a countably closed and a \(\sigma\)-centered part. Then the author uses a mixture of finite and countable supports.

Reviewer: M.Weese

MSC:

03E35	Consistency and independence results
03C62	Models of arithmetic and set theory

Keywords:

iterated forcing; base-matrix tree; distributivity cardinal; tower; model of ZFC; Mathias forcing