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Model theory studies the relationship between formal theories and the structures that satisfy them. A theory T is a set of sentences closed under logical consequence; a model of T is a structure where every sentence in T is true. Two structures are elementarily equivalent if they satisfy exactly the same first-order sentences. A theory is complete if for every sentence φ, either φ or ¬φ is in the theory; it is categorical in cardinality κ if all its models of cardinality κ are isomorphic. These concepts help characterize what first-order logic can and cannot express.
Work through examples of complete theories (dense linear orders without endpoints, algebraically closed fields) and incomplete theories (the theory of groups). Verify elementary equivalence by constructing back-and-forth systems.
From first-order semantics you know how to evaluate a first-order sentence in a given structure M: interpret each constant, function symbol, and relation symbol in M, then check whether the sentence comes out true. Model theory is the systematic study of the relationship between theories — sets of sentences — and the structures that satisfy them.
A theory T is formally a set of first-order sentences closed under logical consequence: whenever T ⊨ φ (every model of T satisfies φ), we require φ ∈ T. A structure M is a model of T if every sentence in T is true in M. The collection of all sentences true in M is its complete theory Th(M) — by definition, a complete theory. Two structures M and N are elementarily equivalent (M ≡ N) if Th(M) = Th(N): they agree on every first-order sentence, even though they may look quite different as structures. For example, the rationals ℚ and the reals ℝ, as ordered fields, are not isomorphic, but they are elementarily equivalent as dense linear orders without endpoints.
A critical distinction: elementary equivalence is much weaker than isomorphism. Isomorphic structures are always elementarily equivalent, but the converse fails dramatically. First-order logic simply cannot distinguish many non-isomorphic structures: for instance, any two algebraically closed fields of the same characteristic and the same uncountable cardinality are isomorphic, but two ACFs of different cardinalities satisfy the same sentences. This gap between "syntactically indistinguishable" and "structurally identical" is one of the central themes of model theory.
A theory T is complete if for every sentence φ, either φ ∈ T or ¬φ ∈ T — the theory has an opinion on every question expressible in the language. Note that completeness is about sentences, not about uniqueness of models: a complete theory can still have many non-isomorphic models, one for each infinite cardinality. The theory DLO (dense linear orders without endpoints) is complete, yet ℚ and ℝ are both models and are non-isomorphic. What completeness buys you is that any two models agree on all first-order sentences — they are elementarily equivalent — even if they are not isomorphic.
Categoricity in cardinality κ goes further: T is κ-categorical if all models of T of cardinality κ are isomorphic to each other. This pins down the structure at size κ completely. The Łoś–Vaught test is a powerful tool: if T is consistent with no finite models and is κ-categorical for some infinite κ, then T is complete. This is how we prove DLO is complete — it is ℵ₀-categorical (every countable dense linear order without endpoints is isomorphic to ℚ) and has no finite models.