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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

Attributes	Values
rdf:type	Thing
rdfs:label	Satz von Morley (Modelltheorie) (de) Categorical theory (en)
rdfs:comment	Der Satz von Morley ist ein Satz aus der Modelltheorie, einem mathematischen Teilgebiet der Logik. Er sagt aus, dass eine abzählbare Theorie, die bis auf Isomorphie in einer überabzählbaren Kardinalzahl nur ein Modell hat, dann auch in jeder überabzählbaren Kardinalzahl nur ein Modell hat. Bewiesen wurde der Satz von Michael D. Morley 1962 in seiner Dissertation Categoricity in Power. Der Satz und sein Beweis übten eine nachhaltige Wirkung auf die Modelltheorie aus. Im Jahr 2003 erhielt Morley dafür den Leroy P. Steele Prize. Saharon Shelah verallgemeinerte 1974 den Satz auf überabzählbare Theorien. (de) 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 (en)
differentFrom	Category theory
dcterms:subject	Theorems in the foundations of mathematics Mathematical logic Model theory
Wikipage page ID	1096396 (xsd:integer)
Wikipage revision ID	1065894592 (xsd:integer)
Link from a Wikipage to another Wikipage	Model theory Omega-categorical theory Algebraically closed field Peano axioms Up to isomorphism Vector space Infinite set Complex numbers Mathematical logic Saharon Shelah Łoś–Vaught test Equivalence class Linear order Löwenheim–Skolem theorem Complete theory Dense order Spectrum of a theory Divisible group Countable American Mathematical Society Equivalence relation Field (mathematics) Finite set First-order logic First-order theory Theorems in the foundations of mathematics Oswald Veblen Cardinal number Cardinality Formal language Graduate Texts in Mathematics Prime number Abelian group Mathematical logic Model theory Characteristic (algebra) Jerzy Łoś Higher-order logic Theory (mathematical logic) Torsion-free abelian group Torsion group Michael D. Morley Natural number Rational number Set (mathematics) Uncountable Transactions of the American Mathematical Society Springer-Verlag P-adic Tarski–Vaught test Model (mathematical logic) Stability (model theory)
Link from a Wikipage to an external page	https://archive.org/details/classificationth0092shel http://plato.stanford.edu/archives/sum2005/entries/modeltheory-fo
sameAs	Categorical theory Categorical theory Categorical theory Categorical theory
dbp:wikiPageUsesTemplate	dbt:Springer dbt:Citation dbt:Distinguish dbt:Math dbt:Reflist dbt:Sfn dbt:Redirect-distinguish2 dbt:Harvs dbt:Mathematical_logic
authorlink	Michael D. Morley (en) Saharon Shelah (en)
first	Michael D. (en) E.A. (en) Saharon (en)
id	c/c020730 (en)
last	Morley (en) Shelah (en) Palyutin (en)
title	Categoricity in cardinality (en)
year	1965 (xsd:integer) 1974 (xsd:integer)
has abstract	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. (en) Der Satz von Morley ist ein Satz aus der Modelltheorie, einem mathematischen Teilgebiet der Logik. Er sagt aus, dass eine abzählbare Theorie, die bis auf Isomorphie in einer überabzählbaren Kardinalzahl nur ein Modell hat, dann auch in jeder überabzählbaren Kardinalzahl nur ein Modell hat. Bewiesen wurde der Satz von Michael D. Morley 1962 in seiner Dissertation Categoricity in Power. Der Satz und sein Beweis übten eine nachhaltige Wirkung auf die Modelltheorie aus. Im Jahr 2003 erhielt Morley dafür den Leroy P. Steele Prize. Saharon Shelah verallgemeinerte 1974 den Satz auf überabzählbare Theorien. (de)
prov:wasDerivedFrom	wikipedia-en:Categorical_theory?oldid=1065894592&ns=0
page length (characters) of wiki page	9922 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Categorical_theory
is Link from a Wikipage to another Wikipage of	Morley categoricity theorem Robert Lawson Vaught Index of philosophy articles (A–C) Glossary of set theory Dense order

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