Consider the sentence: there exists an element that is its own inverse. Written in the formal language of group theory, this statement is perfectly precise. Yet it is true in some groups and false in others. The integers under addition satisfy it — zero is its own inverse. But change the structure, and the sentence's truth value can flip entirely.

This is the central revelation of model theory: formal sentences do not have truth values in isolation. They acquire truth only when interpreted inside a specific mathematical structure. The same string of symbols can be simultaneously true in one world and false in another, and both situations are perfectly legitimate.

Model theory makes this observation rigorous. It gives us the machinery to ask not just what is true, but where it is true, and what that difference tells us about the deep architecture of mathematical theories. What follows is an introduction to that machinery — structures, truth, and the surprising multiplicity of worlds that a single theory can describe.

Every formal language in first-order logic consists of a fixed vocabulary: constant symbols, function symbols, and relation symbols, each with a specified number of argument places. Think of this vocabulary as a script — it tells you the cast of characters and the kinds of interactions that can be discussed, but it says nothing about who those characters actually are.

A first-order structure (also called a model or an interpretation) is what brings the script to life. It consists of three ingredients. First, a non-empty set called the domain or universe — the collection of objects the structure talks about. Second, an assignment of each constant symbol to a specific element of that domain. Third, an assignment of each function symbol to an actual function on the domain, and each relation symbol to an actual relation among domain elements.

For example, the language of ring theory includes symbols for addition, multiplication, a zero constant, and a unity constant. The integers ℤ form one structure interpreting this language: the domain is the set of integers, addition means ordinary addition, and so on. But the set of 2×2 real matrices also forms a structure for the same language — same script, completely different stage. In one structure multiplication is commutative; in the other it is not.

This separation between language and structure is the foundational move of model theory. It forces us to distinguish between what we can say and what happens to be true in a given mathematical world. The language provides the syntax; the structure provides the semantics. Neither is complete without the other, and the interplay between them generates the entire subject.

Takeaway

A formal language is a template with blanks. A structure fills in those blanks with actual mathematical objects. The same template can be filled in radically different ways, and model theory studies exactly what changes and what stays the same across those fillings.

Once a structure is fixed, every sentence of the language receives a definite truth value — true or false — inside that structure. The formal definition proceeds recursively. Atomic sentences like R(c₁, c₂) are true in a structure M precisely when the interpretations of c₁ and c₂ stand in the relation that M assigns to R. Logical connectives — 'and,' 'or,' 'not,' 'implies' — combine truth values in the standard way.

The real subtlety enters with quantifiers. The sentence ∀x P(x) is true in M when every element of M's domain satisfies P. The sentence ∃x P(x) is true when at least one element does. This means that truth under quantifiers depends entirely on what the domain contains. Expand or shrink the domain, and quantified statements can change their truth values even when everything else about the structure stays the same.

We write M ⊨ φ to say that sentence φ is true in structure M, and we call M a model of φ. If T is a set of sentences (a theory), then M ⊨ T means every sentence in T is true in M. This notation — the satisfaction relation — is the backbone of model theory. It transforms vague talk about 'interpretation' into a precise mathematical definition.

Here is the key consequence: a sentence can be logically valid (true in every structure), satisfiable (true in at least one), or unsatisfiable (true in none). Most interesting mathematical statements fall into the middle category — satisfiable but not valid. They carve out a class of structures, and understanding that class is exactly what model theory investigates.

Takeaway

Truth in model theory is always truth relative to a structure. A sentence does not carry its truth value around like a passport — it receives one upon entry into a specific mathematical world. This relativity is not a weakness; it is the engine of the entire theory.

If a first-order theory T is consistent — meaning it has no internal contradictions — then by the completeness theorem, T has at least one model. The Löwenheim-Skolem theorem goes further: if T has an infinite model, then it has models of every infinite cardinality. This is a staggering result. It means that the first-order theory of the real numbers, for instance, has a countable model — a structure with only countably many elements in which every axiom of the theory holds true.

Such models are called non-standard. The standard model of Peano arithmetic is the natural numbers ℕ with their usual addition and multiplication. But Peano arithmetic also has non-standard models containing 'infinite' numbers — elements larger than every standard natural number — that nonetheless satisfy every first-order axiom of arithmetic. These exotic elements are invisible to the axioms; no first-order sentence can distinguish the standard model from a non-standard one.

Two structures are isomorphic if there is a bijection between their domains that preserves all the structure's functions and relations. Non-isomorphic models of the same theory are genuinely different mathematical worlds that happen to agree on every sentence the theory can express. This means first-order theories are inherently unable to pin down a unique structure (up to isomorphism) once infinite domains are involved.

This is not a deficiency of first-order logic — it is a deep structural feature. It tells us something profound about the limits of formal description. No matter how many axioms you write down, if your theory has an infinite model, it has many non-isomorphic models. The gap between what you can say and what you can uniquely determine is an ineradicable part of the logical landscape.

Takeaway

A consistent first-order theory does not describe a single mathematical world — it describes a family of worlds. The inability to uniquely characterize infinite structures with first-order sentences is not a failure of axioms but a fundamental theorem about the expressive limits of formal languages.

Model theory begins with a simple architectural decision: separate the language from its interpretation. Once that separation is in place, truth becomes relative to structure, and the space of possible interpretations opens wide.

The consequences are both precise and philosophically striking. The same axioms that describe the natural numbers also describe strange, bloated cousins containing infinite elements no axiom can rule out. Formal sentences constrain but do not determine.

This is the power — and the humility — of first-order logic. It gives us the most reliable framework for mathematical reasoning ever devised, while honestly revealing the boundaries of what any finite collection of sentences can achieve. Knowing those boundaries is itself a form of certainty.