This page is about the notion of model in logic. For the notion in physics see model (in theoretical physics).
indiscernible sequence?
Morley sequence?
Ramsey theorem?
Erdos-Rado theorem?
Ehrenfeucht-Fraïssé games (back-and-forth games)
Hrushovski construction?
generic predicate?
In model theory, a model of a theory is a realization of the types, operations, relations, and axioms of that theory. In ordinary model theory one usually studies mainly models in sets, but in categorical logic we study models in other categories, especially in topoi.
The term structure is often used to mean a realization of types, operations, and relations in some signature, but not satisfying any particular axioms. This is of course the same as a model for the “empty theory” in that signature, which has the same types, operations, and relations, but no axioms at all. One then talks about whether a given structure is, or is not, a model of a given theory in a given signature.
The basic concept is of a structure for a first-order language : a set together with an interpretation of in . A theory is specified by a language and a set of sentences in .
An -structure is a model of if for every sentence in , its interpretation in , is true (“ holds in ”).
We say that is consistent or satisfiable (relative to the universe in which we do model theory) if there exist at least one model for (in our universe). Two theories, , are said to be equivalent if they have the same models.
Given a class of structures for , there is a theory consisting of all sentences in which hold in every structure from . Two structures and are elementary equivalent (sometimes written by equality , sometimes said “elementarily equivalent”) if , i.e. if they satisfy the same sentences in . Any set of sentences which is equivalent to is called a set of axioms of . A theory is said to be finitely axiomatizable if there exist a finite set of axioms for .
A theory is said to be complete if it is equivalent to for some structure .
For the syntactic category of an algebraic theory (where is perhaps better known as a Lawvere theory), and for any category with finite limits, a model for in is a product-preserving functor
The category of models in this case is hence the full subcategory of the functor category on product-preserving functors.
Last revised on August 10, 2024 at 22:32:11. See the history of this page for a list of all contributions to it.