Kategoriallisuus

Kategoriallisuus, also known as categoricity, is a concept in logic and mathematics that refers to the property of a theory or a formula being categorical. A theory is categorical if it has, up to isomorphism, exactly one model. This means that all models of the theory are essentially the same, differing only in the names of the elements.

In the context of first-order logic, a theory is categorical if it has a model of every infinite cardinality, and all models of the same cardinality are isomorphic. For example, the theory of dense linear orders without endpoints is categorical in the infinite, meaning that any two countably infinite dense linear orders without endpoints are isomorphic.

Categoricity is closely related to the concept of completeness. A theory is complete if it has no non-trivial extensions, meaning that any two models of the theory are elementarily equivalent. A categorical theory is always complete, but the converse is not necessarily true.

The study of categorical theories is important in model theory, as it provides insights into the structure and properties of mathematical systems. Categoricity is also relevant in the philosophy of mathematics, where it is used to argue for the existence of mathematical objects and the uniqueness of mathematical structures.