Model Theory

Model theory is the branch of mathematical logic that studies the relationship between formal languages and their interpretations — the mathematical structures (models) that make the sentences of a language true or false. Where proof theory asks what can be derived from axioms, model theory asks what structures satisfy those axioms. The key result bridging the two is Gödel's Completeness Theorem (distinct from his Incompleteness Theorems): every consistent first-order theory has a model. This means that syntactic consistency and semantic satisfiability coincide for first-order logic — a deep alignment that does not hold for stronger logics. Model theory's most counterintuitive result is the Löwenheim-Skolem theorem: any first-order theory with an infinite model has models of every infinite cardinality. This means that set theory, intended to talk about uncountable infinities, also has countable models — the so-called Skolem paradox, which is not actually a paradox but a reminder that axioms do not uniquely determine their intended interpretation. Non-standard analysis and non-standard arithmetic are among model theory's gifts to mathematics proper.