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 subfield of
mathematical logic
Mathematical logic is the study of logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of for ...
, an atomic model is a model such that the complete
type of every
tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defi ...
is axiomatized by a single
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 ...
. Such types are called principal types, and the formulas that axiomatize them are called complete formulas.
Definitions
Let ''T'' be a
theory
A theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may be s ...
. A complete type ''p''(''x''
1, ..., ''x''
''n'') is called principal or atomic (relative to ''T'') if it is axiomatized relative to ''T'' by a single formula ''φ''(''x''
1, ..., ''x''
''n'') ∈ ''p''(''x''
1, ..., ''x''
''n'').
A formula ''φ'' is called complete in ''T'' if for every formula ''ψ''(''x''
1, ..., ''x''
''n''), the theory ''T'' ∪ entails exactly one of ''ψ'' and ¬''ψ''.
[Some authors refer to complete formulas as "atomic formulas", but this is inconsistent with the purely syntactical notion of an atom or atomic formula as a formula that does not contain a proper subformula.]
It follows that a complete type is principal if and only if it contains a complete formula.
A model ''M'' is called atomic if every ''n''-tuple of elements of ''M'' satisfies a formula that is complete in Th(''M'')—the theory of ''M''.
Examples
*The
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. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered field ...
of
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010)
...
algebraic number
An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of the po ...
s is the unique atomic model of the theory of
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.
*Any finite model is atomic.
*A dense
linear ordering
In mathematics, a total 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 ( reflexi ...
without endpoints is atomic.
*Any
prime model
In mathematics, and in particular model theory, a prime model is a model that is as simple as possible. Specifically, a model P is prime if it admits an elementary embedding into any model M to which it is elementarily equivalent (that is, into an ...
of a
countable
In mathematics, a set is countable if either it is 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 from it into the natural numbers; ...
theory is atomic by the
omitting types theorem
In model theory and related areas of mathematics, a type is an object that describes how a (real or possible) element or finite collection of elements in a structure (mathematical logic), mathematical structure might behave. More precisely, it is ...
.
*Any countable atomic model is prime, but there are plenty of atomic models that are not prime, such as an uncountable dense linear order without endpoints.
*The theory of a countable number of independent unary relations is complete but has no completable formulas and no atomic models.
Properties
The
back-and-forth method In mathematical logic, especially set theory and model theory, the back-and-forth method is a method for showing isomorphism between countably infinite structures satisfying specified conditions. In particular it can be used to prove that
* any t ...
can be used to show that any two countable atomic models of a theory that are
elementarily equivalent In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences.
If ''N'' is a substructure of ''M'', one often ...
are
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word is ...
.
Notes
References
*
*
{{Mathematical logic
Model theory