Back-and-Forth Method

A combinatorial technique for building isomorphisms (or partial isomorphisms and back-and-forth systems) between structures by alternately extending partial maps from each side to ensure element-by-element correspondence.

An inductive technique for constructing an isomorphism between two countable structures (or demonstrating elementary equivalence) by alternately extending a partial isomorphism first from one structure to the other and then back, ensuring coherence at each finite stage.