Weak foundation and axioms of universality.
Weak square sequences and special Aronszajn trees
A classical theorem of set theory is the equivalence of the weak square principle with the existence of a special Aronszajn tree on μ⁺. We introduce the notion of a weak square sequence on any regular uncountable cardinal, and prove that the equivalence between weak square sequences and special Aronszajn trees holds in general.