Mathematical logic | Model theory | Theorems in the foundations of mathematics

Categorical theory

In mathematical logic, a theory is categorical if it has exactly one model (up to isomorphism). Such a theory can be viewed as defining its model, uniquely characterizing its structure. In first-order logic, only theories with a finite model can be categorical. Higher-order logic contains categorical theories with an infinite model. For example, the second-order Peano axioms are categorical, having a unique model whose domain is the set of natural numbers In model theory, the notion of a categorical theory is refined with respect to cardinality. A theory is κ-categorical (or categorical in κ) if it has exactly one model of cardinality κ up to isomorphism. Morley's categoricity theorem is a theorem of Michael D. Morley stating that if a first-order theory in a countable language is categorical in some uncountable cardinality, then it is categorical in all uncountable cardinalities. Saharon Shelah extended Morley's theorem to uncountable languages: if the language has cardinality κ and a theory is categorical in some uncountable cardinal greater than or equal to κ then it is categorical in all cardinalities greater than κ. (Wikipedia).

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Related pages

Algebraically closed field | Vector space | Finite set | Divisible group | Infinite set | Łoś–Vaught test | Löwenheim–Skolem theorem | Spectrum of a theory | Model theory | Theory (mathematical logic) | Rational number | Omega-categorical theory | Torsion-free abelian group | Formal language | Peano axioms | Higher-order logic | Equivalence class | Natural number | Cardinal number | Characteristic (algebra) | Set (mathematics) | Field (mathematics) | Michael D. Morley | Torsion group | Prime number | Complete theory | Dense order | Equivalence relation | Mathematical logic | Cardinality | First-order logic | Abelian group | Oswald Veblen