model theory In mathematical logic, model theory is the study of the relationship between theory (mathematical logic), formal theories (a collection of Sentence (mathematical logic), sentences in a formal language expressing statements about a Structure (mat ...

, a branch of

mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...

, the notion of an existentially closed model (or existentially complete model) of a

theory A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, ...

generalizes the notions of

algebraically closed field In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra ...

s (for the theory of fields),

real closed field In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. Def ...

s (for the theory of

ordered field In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields are the rational numbers and the real numbers, both with their standard ord ...

s), existentially closed groups (for the theory of groups), and

dense Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be use ...

linear order In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation \leq on some set X, which satisfies the following for all a, b and c in X: # a \leq a ( ref ...

s without endpoints (for the theory of linear orders).

Definition

A substructure ''M'' of a

structure A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such as ...

''N'' is said to be existentially closed in (or existentially complete in)

N

if for every quantifier-free

formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...

φ(''x''₁,…,''x''_''n'',''y''₁,…,''y''_''n'') and all elements ''b''₁,…,''b''_''n'' of ''M'' such that φ(''x''₁,…,''x''_''n'',''b''₁,…,''b''_''n'') is realized in ''N'', then φ(''x''₁,…,''x''_''n'',''b''₁,…,''b''_''n'') is also realized in ''M''. In other words: If there is a tuple ''a''₁,…,''a''_''n'' in ''N'' such that φ(''a''₁,…,''a''_''n'',''b''₁,…,''b''_''n'') holds in ''N'', then such a tuple also exists in ''M''. This notion is often denoted

M \prec_1 N

. A model ''M'' of a theory ''T'' is called existentially closed in ''T'' if it is existentially closed in every superstructure ''N'' that is itself a model of ''T''. More generally, a structure ''M'' is called existentially closed in a

class Class, Classes, or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used d ...

''K'' of structures (in which it is contained as a member) if ''M'' is existentially closed in every superstructure ''N'' that is itself a member of ''K''. The existential closure in ''K'' of a member ''M'' of ''K'', when it exists, is, up to

isomorphism In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...

, the least existentially closed superstructure of ''M''. More precisely, it is any extensionally closed superstructure ''M''^∗ of ''M'' such that for every existentially closed superstructure ''N'' of ''M'', ''M''^∗ is isomorphic to a substructure of ''N'' via an isomorphism that is the identity on ''M''.

Examples

Let ''σ'' = (+,×,0,1) be the

signature A signature (; from , "to sign") is a depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. Signatures are often, but not always, Handwriting, handwritt ...

of fields, i.e. + and × are binary function symbols and 0 and 1 are constant symbols. Let ''K'' be the class of structures of signature ''σ'' that are fields. If ''A'' is a subfield of ''B'', then ''A'' is existentially closed in ''B'' if and only if every system of

polynomial In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...

s over ''A'' that has a solution in ''B'' also has a solution in ''A''. It follows that the existentially closed members of ''K'' are exactly the algebraically closed fields. Similarly in the class of

s, the existentially closed structures are the

s. In the class of

s, the existentially closed structures are those that are

without endpoints, while the existential closure of any

countable In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...

(including empty) linear order is, up to isomorphism, the countable dense total order without endpoints, namely the

order type In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y su ...

of the rationals.

References

* * {{Citation , last1=Hodges , first1=Wilfrid , author1-link=Wilfrid Hodges , title=A shorter model theory , publisher=

Cambridge University Press Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessme ...

, location=Cambridge , isbn=978-0-521-58713-6 , year=1997

External links

Encyclopedia of Mathematics article
Model theory

Definition

Examples

See also

References

External links