Aronszajn line

In mathematical set theory, an Aronszajn line is a linear ordering of cardinality
which contains no subset order-isomorphic to

with the usual ordering
the reverse of
an uncountable subset of the Real numbers with the usual ordering.

Unlike Suslin lines, the existence of Aronszajn lines is provable using the standard axioms of set theory. A linear ordering is an Aronszajn line if and only if it is the lexicographical ordering of some Aronszajn tree.