In
model theory
In mathematical logic, 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 (those structures in which the s ...
, a branch of
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an imaginary element of a structure is roughly a definable
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
. These were introduced by , and elimination of imaginaries was introduced by .
Definitions
*''M'' is a
model of some
theory.
*x and y stand for ''n''-tuples of variables, for some
natural number ''n''.
*An equivalence formula is a
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 betwee ...
φ(x, y) that is a
symmetric and
transitive relation
Relation or relations may refer to:
General uses
*International relations, the study of interconnection of politics, economics, and law on a global level
*Interpersonal relationship, association or acquaintance between two or more people
*Public ...
. Its domain is the
set of elements a of ''M''
''n'' such that φ(a, a); it is an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
on its domain.
*An imaginary element a/φ of ''M'' is an equivalence formula φ together with an equivalence class a.
*''M'' has elimination of imaginaries if for every imaginary element a/φ there is a formula θ(x, y) such that there is a unique tuple b so that the equivalence class of a consists of the tuples x such that θ(x, b).
*A model has uniform elimination of imaginaries if the formula θ can be chosen independently of a.
*A theory has elimination of imaginaries if every model of that theory does (and similarly for uniform elimination).
Examples
*
ZFC set theory has elimination of imaginaries.
*
Peano arithmetic
In mathematical logic, the Peano axioms, also known as the Dedekind–Peano axioms or the Peano postulates, are axioms for the natural numbers presented by the 19th century Italian mathematician Giuseppe Peano. These axioms have been used nearly u ...
has uniform elimination of imaginaries.
*A
vector space of
dimension at least 2 over a
finite field with at least 3 elements does not have elimination of imaginaries.
References
*
*
*{{Citation , last1=Shelah , first1=Saharon , author1-link=Saharon Shelah , title=Classification theory and the number of nonisomorphic models , origyear=1978 , publisher=Elsevier , edition=2nd , series=Studies in Logic and the Foundations of Mathematics , isbn=978-0-444-70260-9 , year=1990 , url-access=registration , url=https://archive.org/details/classificationth0092shel
Model theory