Elementary Embedding
In model theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ..., a branch of mathematical logic Mathematical logic, also called formal logic, is a subfield of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry) ..., two structure A structure is an arrangement and organization of interrelated elements in a material object or system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A sy ...s ''M'' and ''N'' of the same signature 's signature is the most prominent on the United States Declaration of Independence and the Articles of Confederation. The name "John ... [...More Info...] [...Related Items...] 

Model Theory
In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It has no generally ..., model theory is the study of the relationship between formal theories (a collection of sentences in a formal language expressing statements about a mathematical structure), and their models, taken as Interpretation (model theory), interpretations that satisfy the sentences of that theory. Informal description Model theory recognizes and is intimately concerned with a duality: it examines Semantics, semantical elements (meaning and truth) by means of Syntax, syntactical elements (formulas and proofs) of a corresponding language. In a summary definition, dating from 1973: :model theory = universal algebra + logic. Model theory developed rapidly during the 1990s, and a more modern definition is provided by Wilfrid ... [...More Info...] [...Related Items...] 

Linear Ordering
In mathematics, a total order, simple order, linear order, connex order, or full order is a binary relation on some Set (mathematics), set X, which is Antisymmetric relation, antisymmetric, Transitive relation, transitive, and a connex relation. A set paired with a total order is called a chain, a totally ordered set, a simply ordered set, a linearly ordered set, or a loset. Formally, a binary relation \leq is a total order on a set X if the following statements hold for all a, b and c in X: : Antisymmetric relation, Antisymmetry: If a \leq b and b \leq a then a = b; : Transitive relation, Transitivity: If a \leq b and b \leq c then a \leq c; : Connex relation, Connexity: a \leq b or b \leq a. Antisymmetry eliminates uncertain cases when both a precedes b and b precedes a. A relation having the ''connex'' property means that any pair of elements in the set of the relation are comparability, comparable under the relation. This also means that the set can be diagrammed as a line ... [...More Info...] [...Related Items...] 