Model theory studies the relationship between formal logical languages and the mathematical structures — called models — that make the statements in those languages true or false. A single set of axioms can have many genuinely different models, and understanding how the language and its models relate has led to deep and surprising applications throughout algebra, number theory, and geometry, far beyond its foundational origins.
Consider the axioms of a group (see the Group entry): a set of rules about an operation (closure, associativity, identity, inverses). Those rules are satisfied by the integers under addition, by the symmetries of a square, by the rotations of a sphere, and by countless other mathematical systems — all are "models" of the same group axioms, yet they are completely different objects with different sizes, structures, and behaviours. Model theory makes this observation the start of a deep, systematic investigation: how does the choice of axioms constrain the models? What can the language express? What can it never distinguish?
The Peano axioms for arithmetic — our standard rules for the natural numbers — have a "standard" model: the familiar natural numbers 0, 1, 2, 3, .... But they also have infinitely many "non-standard" models containing elements that behave like infinitely large natural numbers, satisfying every first-order statement that ordinary natural numbers satisfy, yet genuinely different. This initially startling fact (provable rigorously from the compactness theorem — see below) reveals a fundamental gap between what first-order language can express and the full concept of "the natural numbers."
If a set of first-order axioms has any infinite model at all, then it has models of every infinite cardinality — both larger and smaller (down to countably infinite). This striking result, due to Leopold Löwenheim (1915) and Thoralf Skolem (1920), means that first-order axioms can never uniquely characterise a structure by specifying its size. Even the axioms for the real numbers — which intend to describe an uncountable structure — have a countable model. This apparent paradox (sometimes called Skolem's paradox) reveals deep limitations of first-order languages.
A set of first-order sentences has a model if and only if every finite subset of it has a model. This elegant result — the compactness theorem — is one of model theory's most powerful tools. It implies, for example, that the Peano axioms have non-standard models (by adding sentences asserting the existence of an element larger than every standard natural number, and observing that every finite subset of this enlarged axiom set is satisfiable), and it underlies many existence proofs throughout model theory.
Two structures are elementarily equivalent if they satisfy exactly the same first-order sentences — even if they are not isomorphic (not the same structure in a stronger sense). Elementary equivalence captures what is expressible in first-order language, while isomorphism captures full structural identity. The gap between these two notions is a central theme of model theory.
A theory is κ-categorical (for a cardinal κ) if all its models of size κ are isomorphic to each other — the theory pins down a unique structure of that size. Morley's categoricity theorem (1965) — a landmark result — proves that if a countable first-order theory is categorical in one uncountable cardinal, it is categorical in all uncountable cardinals. This theorem launched the modern study of stability theory.
Saharon Shelah's monumental stability theory (developed from the early 1970s, culminating in his 1978 book Classification Theory) asks: for a given first-order theory, how many non-isomorphic models of each infinite cardinality does it have? Shelah showed that theories fall into a precise classification — stable, superstable, ω-stable, and others — based on this counting function, and that stable theories have far more tractable structure theory than unstable ones. This programme, lasting decades and involving enormous combinatorial and set-theoretic machinery, is one of the most ambitious and sustained research projects in 20th-century pure mathematics.
O-minimality (developed by Lou van den Dries and Alex Wilkie in the 1980s–1990s) identifies a class of structures — "o-minimal structures" — with particularly tame definable sets, generalising the well-behaved geometry of semialgebraic sets over the reals. O-minimal structures have proven remarkably useful in attacking problems in real algebraic geometry, analytic geometry, and even number theory — including contributions to the André-Oort conjecture, a major open problem connecting model theory and arithmetic geometry.
Model theory's most striking modern development may be its unexpected effectiveness in pure algebra and number theory. Ax-Kochen theorem (1965): using model-theoretic tools, James Ax and Simon Kochen proved a result about the solvability of polynomial equations over p-adic fields that had resisted purely algebraic approaches. Hrushovski's theorem (1996): Ehud Hrushovski used model-theoretic methods (specifically, geometric stability theory) to prove the Mordell-Lang conjecture for function fields — a major result in arithmetic geometry. These applications, striking to both logicians and algebraists when they first appeared, established model theory as a genuinely powerful tool in mainstream mathematics.