Type (model theory)

In <a href="/facts/Model_theory/vLWVV8sQ">model theory</a> and related areas of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, a type is an object that describes how a (real or possible) element or finite collection of elements in a <a href="/facts/Structure_(mathematical_logic)/GsMx1fEb">mathematical structure</a> might behave. More precisely, it is a set of <a href="/facts/First-order_logic/lcISqWIg">first-order</a> formulas in a language L with <a href="/facts/Free_variable/ndWnXXYs">free variables</a> x1, x2,..., xn that are true of a set of <a href="/facts/N-tuple/BI1HIxL8">n-tuples</a> of an L-structure 
 
 
 
 
 
 M
 
 
 
 
 {\displaystyle {\mathcal {M}}}
 
. Depending on the context, types can be complete or partial and they may use a fixed set of constants, A, from the structure 
 
 
 
 
 
 M
 
 
 
 
 {\displaystyle {\mathcal {M}}}
 
. The question of which types represent actual elements of 
 
 
 
 
 
 M
 
 
 
 
 {\displaystyle {\mathcal {M}}}
 
 leads to the ideas of <a href="/facts/Saturated_model/FvPcXTSr">saturated models</a> and omitting types.

Type (model theory) open-in-new

Type (model theory)