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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:
 3e+35 Consistency and independence results 300000 Other combinatorial set theory
##### Keywords:
special tree; forcing; consistency; Aronszajn tree
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