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 ...
and related areas of
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 ...
, a type is an object that describes how a (real or possible) element or finite collection of elements in a
In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology). Often, the additional features are attached or related to the set, so as to provide it with some additi ...
might behave. More precisely, it is a set of
In mathematics and other formal sciences, first-order or first order most often means either:
* "linear" (a polynomial of degree at most one), as in first-order approximation and other calculus uses, where it is contrasted with "polynomials of hig ...
formulas in a language ''L'' with free variables ''x''1
that are true of a sequence of elements of an ''L''-structure
. Depending on the context, types can be complete or partial and they may use a fixed set of constants, ''A'', from the structure
. The question of which types represent actual elements of
leads to the ideas of saturated model
s and omitting types.
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 ...
Language is a structured system of communication. The structure of a language is its grammar and the free components are its vocabulary. Languages are the primary means by which humans communicate, and may be conveyed through a variety of ...
''L''. Let ''M'' be the
The universe is all of space and time and their contents, including planets, stars, galaxies, and all other forms of matter and energy. The Big Bang theory is the prevailing cosmological description of the development of the univer ...
of the structure. For every ''A'' ⊆ ''M'', let ''L''(''A'') be the language obtained from ''L'' by adding a constant ''c''''a''
for every ''a'' ∈ ''A''. In other words,
A 1-type (of
) over ''A'' is a set ''p''(''x'') of formulas in ''L''(''A'') with at most one free variable ''x'' (therefore 1-type) such that for every finite subset ''p''0
(''x'') ⊆ ''p''(''x'') there is some ''b'' ∈ ''M'', depending on ''p''0
(i.e. all formulas in ''p''0
(''x'') are true in
when ''x'' is replaced by ''b'').
Similarly an ''n''-type (of
) over ''A'' is defined to be a set ''p''(''x''1
) = ''p''(''x'') of formulas in ''L''(''A''), each having its free variables occurring only among the given ''n'' free variables ''x''1
, such that for every finite subset ''p''0
(''x'') ⊆ ''p''(''x'') there are some elements ''b''1
∈ ''M'' with
A complete type of
over ''A'' is one that is maximal
with respect to inclusion. Equivalently, for every
. Any non-complete type is called a partial type.
So, the word type in general refers to any ''n''-type, partial or complete, over any chosen set of parameters (possibly the empty set).
An ''n''-type ''p''(''x'') is said to be realized in
if there is an element ''b'' ∈ ''M''''n''
. The existence of such a realization is guaranteed for any type by the compactness theorem
, although the realization might take place in some elementary extension
, rather than in
If a complete type is realized by ''b'' in
, then the type is typically denoted
and referred to as the complete type of ''b'' over ''A''.
A type ''p''(''x'') is said to be isolated by ''
, if for all
. Since finite subsets of a type are always realized in
, there is always an element ''b'' ∈ ''M''''n''
such that ''φ''(''b'') is true in
, thus ''b'' realizes the entire isolated type. So isolated types will be realized in every elementary substructure or extension. Because of this, isolated types can never be omitted (see below).
A model that realizes the maximum possible variety of types is called a saturated model
, and the ultrapower
construction provides one way of producing saturated models.
Examples of types
Consider the language with one binary connective, which we denote as
be the structure
for this language, which is the ordinal
with its standard well-ordering. Let
denote the theory of
Consider the set of formulas
. First, we claim this is a type. Let
be a finite subset of
. We need to find a
that satisfies all the formulas in
. Well, we can just take the successor of the largest ordinal mentioned in the set of formulas
. Then this will clearly contain all the ordinals mentioned in
. Thus we have that
is a type.
Next, note that
is not realized in
. For, if it were there would be some
that contains every element of
If we wanted to realize the type, we might be tempted to consider the model
, which is indeed a supermodel of
that realizes the type. Unfortunately, this extension is not elementary, that is, this model does not have to satisfy
. In particular, the sentence
is satisfied by this model and not by
So, we wish to realize the type in an elementary extension. We can do this by defining a new structure in the language, which we will denote
. The domain of the structure will be
is the set of integers adorned in such a way that
denote the usual order of
. We interpret the symbol
in our new structure by
. The idea being that we are adding a "
-chain", or copy of the integers, above all the finite ordinals. Clearly any element of
realizes the type
. Moreover, one can verify that this extension is elementary.
Another example: the complete type of the number 2 over the empty set, considered as a member of the natural numbers, would be the set of all first-order statements, describing a variable ''x'', that are true when ''x'' = 2. This set would include formulas such as