Order automorphism ($\mathrm{Aut}(I,<)$)

An order automorphism of a total order $(I,<)$ is a bijection $\pi :I\to I$ such that $i<j$ if and only if $\pi (i)<\pi (j)$. Composition and inverses are again order automorphisms. Transporting the indices of an order-indiscernible sequence by such a map preserves every first-order formula on increasing finite tuples.

An order-dense subset with no nonidentity order automorphism. A transfinite diagonal construction marks an included point and an excluded image for every nonidentity real-line order automorphism, yielding such a subset of size $2^{{\aleph }_0}$.

An order automorphism of an order-dense subset $D$ of the real line extends uniquely by $\tilde{\phi }(x)=\sup \{\phi (d):d\in D,d<x\}$. Order completeness supplies the extension, and the lower cuts in $D$ give uniqueness.