Aronszajn line

In mathematical set theory, an Aronszajn line (named after Nachman Aronszajn) is a linear ordering of cardinality $\backslash aleph\_1$

which contains no subset order-isomorphic to

• $\backslash omega\_1$ with the usual ordering
• the reverse of $\backslash omega\_1$
• 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.{{cite journal

|title=Lexicographically ordered trees

|last1=Funk | first1=Will

|last2=Lutzer | first2=David J.

|journal=Topology and Its Applications

|volume=152

|date=2005

|issue=3

|pages=275–300

|doi=10.1016/j.topol.2004.10.011

|zbl=1071.03032

|doi-access=

}}