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A non-special \(\omega_ 2\)-tree with special \(\omega_ 1\)-subtrees. (English) Zbl 0711.03022
A tree T is a \(\kappa\)-tree if both the cardinality and the height of T are \(\kappa\). A \(\kappa\)-tree is \(\kappa\)-Aronszajn iff it does not have \(\kappa\)-branches and its levels have cardinalities \(<\kappa\). A \(\kappa\)-tree T is special if there is a function f: \(T\to \beta <\kappa\) such that for all x,y,z\(\in T\) with \(f(x)=f(y)=f(z)\), if \(x<y\) and \(x<z\), then y and z are comparable. The author shows that if ZF is consistent then so is ZFC \(+\) GCH \(+\) “there exists a non-special \(\omega_ 2\)- Aronszajn tree having only special \(\omega_ 1\)-subtrees”.
Reviewer: M.Weese

MSC:
03E35 Consistency and independence results
03E05 Other combinatorial set theory
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