= Branching form of the tree property
For an <uncountable> cardinal $\kappa$, this convention requires a cofinal branch in every separated rooted tree of height $\kappa$ with fewer than $\kappa$ immediate successors per node, without bounding level sizes. It is stronger than the usual <tree property> away from inaccessible cardinals. If $\operatorname{cf}(\kappa)<\kappa$, attach chains of cofinally increasing lengths below $\kappa$ above fewer than $\kappa$ children of one root; this contradicts the property. If $2^\lambda\geq\kappa$ for $\lambda<\kappa$, take a binary tree through level $\lambda$, choose $\kappa$ distinct nodes there, and attach to the $\xi$th a chain of length $\xi$. It has height $\kappa$, binary branching, and no cofinal branch. Hence the property implies that $\kappa$ is a <strongly inaccessible cardinal>. At such a cardinal a separated tree with small immediate branching has small levels, so this convention agrees with the usual <tree property>. It is not the different notion called the strong tree property in modern <set> theory.
Back to article page
