Search results
Results From The WOW.Com Content Network
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 ( 1965 ) stating that if a first-order theory in a countable language is categorical in some uncountable cardinality , then it is categorical in all uncountable ...
A theory T is called κ-categorical if any two models of T that are of cardinality κ are isomorphic. It turns out that the question of κ -categoricity depends critically on whether κ is bigger than the cardinality of the language (i.e. ℵ 0 + | σ | {\displaystyle \aleph _{0}+|\sigma |} , where | σ | is the cardinality of the signature).
Categorical semantics Categorical logic introduces the notion of structure valued in a category C with the classical model theoretic notion of a structure appearing in the particular case where C is the category of sets and functions. This notion has proven useful when the set-theoretic notion of a model lacks generality and/or is inconvenient.
Categorical distribution, general model; Chi-squared test; Cochran–Armitage test for trend; Cochran–Mantel–Haenszel statistics; Correspondence analysis; Cronbach's alpha; Diagnostic odds ratio; G-test; Generalized estimating equations; Generalized linear models; Krichevsky–Trofimov estimator; Kuder–Richardson Formula 20; Linear ...
The theory of random graphs is ω categorical, complete, and decidable, and its countable model is called the Rado graph. A statement in the language of graphs is true in this theory if and only if the probability that an n -vertex random graph models the statement tends to 1 in the limit as n goes to infinity.
In mathematical logic, an omega-categorical theory is a theory that has exactly one countably infinite model up to isomorphism. Omega-categoricity is the special case κ = ℵ 0 {\displaystyle \aleph _{0}} = ω of κ-categoricity , and omega-categorical theories are also referred to as ω-categorical .
Examples: the theory of the set ω×n acted on by the wreath product of G with all permutations of ω.. Examples: theories that are categorical in uncountable cardinals, such as the theory of algebraically closed fields in a given characteristic.. Examples: theories with a finite model, and the inconsistent theory.
Model categories can provide a natural setting for homotopy theory: the category of topological spaces is a model category, with the homotopy corresponding to the usual theory. Similarly, objects that are thought of as spaces often admit a model category structure, such as the category of simplicial sets .