= Solution
\b[A <well-pruned set-theoretic tree> of height $\kappa$ has the property that every node $t$ extends to every higher level below $\kappa$: if $\operatorname{ht}(t)<\beta<\kappa$, there is $u$ of height $\beta$ with $t<_T u$.] Equivalently, the heights of extensions of every node are unbounded in $\kappa$, since taking predecessors then gives an extension at any prescribed intermediate level. This is stronger than merely having no terminal nodes. For a <kappa-tree> we use the usual regular uncountable height <cardinal> and levels of size less than that <cardinal>.
Back to article page
