Ehrenfeucht–Mostowski theorem

Let $M$ be an infinite first-order structure and $I$ a total order. After choosing a Skolem expansion of $M$, an elementary extension contains distinct elements $(a_i{)}_{i\in I}$ forming an order-indiscernible sequence in that expanded language. Their Skolem hull is an elementary substructure in the expanded language, and its reduct is a first-order model of $\mathrm{Th}(M)$. Every order automorphism of $I$ extends uniquely to a structure automorphism of the hull preserving the chosen Skolem expansion, by transporting terms in the generators.

Uniqueness need not hold among all structure automorphisms of the reduct. For an infinite structure in the pure logical equality language, take two distinct nullary Skolem function values $c_0,c_1$ in the hull. For an infinite order-indiscernible sequence, neither can equal a generator: the expanded formula $x=c_j$ would otherwise hold at every generator, contradicting their distinctness. Swapping $c_0,c_1$ fixes every generator and preserves the pure logical equality reduct, while failing to preserve the Skolem expansion. Thus the identity order automorphism already has two extensions in that reduct.