Aronszajn line

In mathematical set theory, an Aronszajn line (named after Nachman Aronszajn) is a linear ordering of cardinality ${\mathrm{\aleph }}_1$ which contains no subset order-isomorphic to

• ${\omega }_1$ with the usual ordering
• the reverse of ${\omega }_1$
• 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.

References

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