Kappa-tree ($\kappa \text{-tree}$)

For a regular uncountable cardinal number $\kappa$, a set-theoretic tree of height $\kappa$ with nonempty levels of cardinality less than $\kappa$. The regular-height convention is important for the unbounded-extension kernel of a regular tree.

For an infinite cardinal $\kappa$, the tree property says that every tree of height $\kappa$ with nonempty levels of cardinality less than $\kappa$ has a cofinal branch. In the all-trees-of-size-$\kappa$ convention it also forces regularity: disjoint chains of lengths cofinal in a singular $\kappa$ form a counterexample. It does not, by itself, imply strong inaccessibility. A strong limit cardinal with the tree property satisfies the self-partition relation by applying it to the end-homogeneous coloring tree. At a strongly inaccessible cardinal this is the usual weak-compactness characterization.

For infinite regular cardinals $\kappa <\lambda$, a $\lambda$-tree whose levels all have size less than $\kappa$ has a cofinal branch. At levels of cofinality $\kappa$, bound all heights distinguishing distinct predecessor chains. The Fodor lemma makes this bound constant on a stationary set; one predecessor at that height occurs stationarily often. The resulting predecessor chains agree below their common heights and unite into a branch through every level. Distinct limit-level nodes with identical predecessor chains do not affect this argument.