A linearly ordered family (M_i : i ∈ I) of structures in a fixed language such that for each i

Principle

Build larger models by successively extending along elementary embeddings so that truth of first-order formulas is preserved at each stage and the union remains an elementary extension of each member.

Demonstration

Given a chain (M_n)_{n∈ω} where M_n ⊨ T and M_n ≺ M_{n+1}, the union ⋃_{n} M_n is a model of T and is an elementary extension of every M_n; this technique constructs countable elementary extensions or direct limits in model theory.

Misapplication

Assuming unions of arbitrarily indexed unions without checking directedness or elementarity will preserve theory; or treating mere inclusion chains (not elementary) as elementary chains, which can break preservation of formulas.

Consequence

Chains let one obtain limit models, control saturation by approximations, and prove existence results (e.g., saturated models via unions of increasing chains of elementary extensions).

Reversal

Working with non-elementary chains or arbitrary directed systems of embeddings where formulas are not preserved pointwise, forcing re-verification of truth at the limit.

Boundary

An elementary chain requires elementarity at successor steps; arbitrary increasing sequences of substructures that are not elementary embeddings fall outside the notion; the property is first-order truth-preserving, not merely isomorphic inclusion.

Semantic Tension

The phrase competes with the informal idea of any increasing sequence of structures; the technical demand of elementary embedding distinguishes it from ordinary chains of substructures.

Synthesis

An elementary chain is an increasing, linearly ordered family of structures linked by elementary embeddings so that their union behaves as a coherent elementary extension preserving first-order truth.