A Tree Axiom

Đuro Kurepa

In connection with my previous results from 1935, and results of other mathematicians (Tarski, Erdös, Hanf, Keisler, Baumgartger$\ldots $) the following Tree (or Dendrity) Axiom is formulated: For any regular uncountable ordinal $n$ there exists a tree An of height (rank) $n$ such that $|X|< |n|$ for every level $X$ as well as for every subchain $X$ of An. In other words, the following assertion Dn holds: There exists a tree $T$ such that for every regular ordinal $n>\omega_0$ the conditions (2:0), (2:1), (2:2) hold.