Inmathematical logic, atheory iscategorical if it has exactly onemodel (up to isomorphism).^[a] Such a theory can be viewed asdefining its model, uniquely characterizing the model's structure.

Infirst-order logic, only theories with afinite model can be categorical.Higher-order logic contains categorical theories with aninfinite model. For example, the second-orderPeano axioms are categorical, having a unique model whose domain is theset of natural numbers${\displaystyle \mathbb{N}.}$

Inmodel theory, the notion of a categorical theory is refined with respect tocardinality. A theory isκ-categorical (orcategorical inκ) if it has exactly one model of cardinalityκ up to isomorphism.Morley's categoricity theorem is a theorem ofMichael D. Morley (1965) stating that if afirst-order theory in acountable language is categorical in someuncountablecardinality, then it is categorical in all uncountable cardinalities.

Saharon Shelah (1974) extended Morley's theorem to uncountable languages: if the language has cardinalityκ and a theory is categorical in some uncountable cardinal greater than or equal toκ then it is categorical in all cardinalities greater than κ.

Oswald Veblen in 1904 defined a theory to becategorical if all of its models are isomorphic. It follows from the definition above and theLöwenheim–Skolem theorem that anyfirst-order theory with a model of infinitecardinality cannot be categorical. One is then immediately led to the more subtle notion ofκ-categoricity, which asks: for which cardinalsκ is there exactly one model of cardinalityκ of the given theoryT up to isomorphism? This is a deep question and significant progress was only made in 1954 whenJerzy Łoś noticed that, at least forcomplete theoriesT over countablelanguages with at least one infinite model, he could only find three ways forT to beκ-categorical at some κ:

• T istotally categorical,i.e.T isκ-categorical for all infinitecardinals κ.
• T isuncountably categorical,i.e.T isκ-categorical if and only ifκ is anuncountable cardinal.
• T iscountably categorical,i.e.T isκ-categorical if and only ifκ is a countable cardinal.

In other words, he observed that, in all the cases he could think of,κ-categoricity at any one uncountable cardinal impliedκ-categoricity at all other uncountable cardinals. This observation spurred a great amount of research into the 1960s, eventually culminating inMichael Morley's famous result that these are in fact the only possibilities. The theory was subsequently extended and refined bySaharon Shelah in the 1970s and beyond, leading tostability theory and Shelah's more general programme ofclassification theory.

There are not many natural examples of theories that are categorical in some uncountable cardinal. The known examples include:

• Pure identity theory (with no functions, constants, predicates other than "=", or axioms).
• The classic example is the theory ofalgebraically closedfields of a givencharacteristic. Categoricity doesnot say that all algebraically closed fields of characteristic 0 as large as thecomplex numbersC are the same asC; it only asserts that they are isomorphicas fields toC. It follows that although the completedp-adic closuresC_p are all isomorphic as fields toC, they may (and in fact do) have completely differenttopological and analytic properties. The theory of algebraically closed fields of a given characteristic isnot categorical inω (the countable infinite cardinal); there are models oftranscendence degree 0, 1, 2, ...,ω.
• Vector spaces over a given countable field. This includesabelian groups of givenprimeexponent (essentially the same as vector spaces over a finite field) anddivisibletorsion-free abelian groups (essentially the same as vector spaces over therationals).
• The theory of the set ofnatural numbers with a successor function.

There are also examples of theories that are categorical inω but not categorical in uncountable cardinals. The simplest example is the theory of anequivalence relation with exactly twoequivalence classes, both of which are infinite. Another example is the theory ofdenselinear orders with no endpoints;Cantor proved that any such countable linear order is isomorphic to the rational numbers: seeCantor's isomorphism theorem.

Every categorical theory iscomplete.^[1] However, the converse does not hold.^[2]

Any theoryT categorical in some infinite cardinalκ is very close to being complete. More precisely, theŁoś–Vaught test states that if a satisfiable theory has no finite models and is categorical in some infinite cardinalκ at least equal to the cardinality of its language, then the theory is complete. The reason is that all infinite models are first-order equivalent to some model of cardinalκ by theLöwenheim–Skolem theorem, and so are all equivalent as the theory is categorical inκ. Therefore, the theory is complete as all models are equivalent. The assumption that the theory have no finite models is necessary.^[3]

• Spectrum of a theory

• Chang, Chen Chung;Keisler, H. Jerome (1990) [1973],Model Theory, Studies in Logic and the Foundations of Mathematics, Elsevier,ISBN 978-0-444-88054-3
• Corcoran, John (1980), "Categoricity",History and Philosophy of Logic,1 (1–2):187–207,doi:10.1080/01445348008837010
• Hodges, Wilfrid, "First-order Model Theory", The Stanford Encyclopedia of Philosophy (Summer 2005 Edition), Edward N. Zalta (ed.).
• Marker, David (2002),Model theory: An introduction,Graduate Texts in Mathematics, vol. 217, New York, NY:Springer-Verlag,ISBN 0-387-98760-6,Zbl 1003.03034
• Monk, J. Donald (1976),Mathematical Logic, Springer-Verlag,doi:10.1007/978-1-4684-9452-5,ISBN 978-1-4684-9454-9
• Morley, Michael (1965), "Categoricity in Power",Transactions of the American Mathematical Society,114 (2),American Mathematical Society, Vol. 114, No. 2:514–538,doi:10.2307/1994188,ISSN 0002-9947,JSTOR 1994188
• Palyutin, E.A. (2001) [1994],"Categoricity in cardinality",Encyclopedia of Mathematics,EMS Press
• Shelah, Saharon (1974), "Categoricity of uncountable theories",Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. of California, Berkeley, Calif., 1971), Proceedings of Symposia in Pure Mathematics, vol. 25, Providence, R.I.: American Mathematical Society, pp. 187–203,doi:10.1090/pspum/025/0373874,ISBN 9780821814253,MR 0373874
• Shelah, Saharon (1990) [1978],Classification theory and the number of nonisomorphic models, Studies in Logic and the Foundations of Mathematics (2nd ed.), Elsevier,ISBN 978-0-444-70260-9 (IX, 1.19, pg.49)
• Veblen, Oswald (1904), "A System of Axioms for Geometry",Transactions of the American Mathematical Society,5 (3), American Mathematical Society, Vol. 5, No. 3:343–384,doi:10.2307/1986462,ISSN 0002-9947,JSTOR 1986462

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

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