Categorical theory

In

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 ${\displaystyle \mathbb {N} .}$

In

Saharon Shelah (1974) 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.

• T is totally categorical, i.e. T is κ-categorical for all infinite cardinals κ.
• T is uncountably categorical, i.e. T is κ-categorical if and only if κ is an
  uncountable
  cardinal.
• T is countably categorical, i.e. T is κ-categorical if and only if κ 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 implied κ-categoricity at all other uncountable cardinals. This observation spurred a great amount of research into the 1960s, eventually culminating in

• Pure identity theory (with no functions, constants, predicates other than "=", or axioms).
• The classic example is the theory of
  transcendence degree
  0, 1, 2, ..., ω.
• Vector spaces over a given countable field. This includes abelian groups of given prime exponent (essentially the same as vector spaces over a finite field) and divisible torsion-free abelian groups (essentially the same as vector spaces over the rationals).
• The theory of the set of natural numbers with a successor function.

There are also examples of theories that are categorical in ω but not categorical in uncountable cardinals. The simplest example is the theory of an

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

Any theory T categorical in some infinite cardinal κ is very close to being complete. More precisely, the Łoś–Vaught test states that if a satisfiable theory has no finite models and is categorical in some infinite cardinal κ 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 cardinal κ by the Löwenheim–Skolem theorem, and so are all equivalent as the theory is categorical in κ. 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

• ISBN 978-0-444-88054-3
• 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,
  Zbl 1003.03034
• Monk, J. Donald (1976), Mathematical Logic, Springer-Verlag,
  doi:10.1007/978-1-4684-9452-5
• JSTOR 1994188
• Palyutin, E.A. (2001) [1994], "Categoricity in cardinality", Encyclopedia of Mathematics, EMS Press
• MR 0373874
• ISBN 978-0-444-70260-9
  (IX, 1.19, pg.49)
• JSTOR 1986462