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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 ...
and related areas 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 ...
, a type is an object that describes how a (real or possible) element or finite collection of elements in a
mathematical structure 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
first-order 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, ''x''2,…, ''x''''n'' that are true of a sequence of elements of an ''L''-structure $\mathcal$. Depending on the context, types can be complete or partial and they may use a fixed set of constants, ''A'', from the structure $\mathcal$. The question of which types represent actual elements of $\mathcal$ leads to the ideas of saturated models and omitting types.

# Formal definition

Consider 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 ...
$\mathcal$ for a
language 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
universe 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, :$L\left(A\right) = L \cup \.$ A 1-type (of $\mathcal$) 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(''x''), with $\mathcal \models p_0\left(b\right)$ (i.e. all formulas in ''p''0(''x'') are true in $\mathcal$ when ''x'' is replaced by ''b''). Similarly an ''n''-type (of $\mathcal$) over ''A'' is defined to be a set ''p''(''x''1,…,''x''''n'') = ''p''(''x'') of formulas in ''L''(''A''), each having its free variables occurring only among the given ''n'' free variables ''x''1,…,''x''''n'', such that for every finite subset ''p''0(''x'') ⊆ ''p''(''x'') there are some elements ''b''1,…,''b''''n'' ∈ ''M'' with $\mathcal\models p_0\left(b_1,\ldots,b_n\right)$. A complete type of $\mathcal$ over ''A'' is one that is maximal with respect to inclusion. Equivalently, for every $\phi\left(\boldsymbol\right) \in L\left(A,\boldsymbol\right)$ either $\phi\left(\boldsymbol\right) \in p\left(\boldsymbol\right)$ or $\lnot\phi\left(\boldsymbol\right) \in p\left(\boldsymbol\right)$. 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 $\mathcal$ if there is an element ''b'' ∈ ''M''''n'' such that $\mathcal\models p\left(\boldsymbol\right)$. 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 of $\mathcal$, rather than in $\mathcal$ itself. If a complete type is realized by ''b'' in $\mathcal$, then the type is typically denoted $tp_n^\left(\boldsymbol/A\right)$ and referred to as the complete type of ''b'' over ''A''. A type ''p''(''x'') is said to be isolated by ''$\varphi$'', for $\varphi \in p\left(x\right)$, if for all $\psi\left(\boldsymbol\right) \in p\left(\boldsymbol\right),$ we have $\operatorname\left(\mathcal M\right) \models \varphi\left(\boldsymbol\right) \rightarrow \psi\left(\boldsymbol\right)$. Since finite subsets of a type are always realized in $\mathcal$, there is always an element ''b'' ∈ ''M''''n'' such that ''φ''(''b'') is true in $\mathcal$; i.e. $\mathcal \models \varphi\left(\boldsymbol\right)$, 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 $\in$. Let $\mathcal$ be the structure $\langle \omega, \in_\rangle$ for this language, which is the ordinal $\omega$ with its standard well-ordering. Let $\mathcal$ denote the theory of $\mathcal$. Consider the set of formulas $p\left(x\right):=\$. First, we claim this is a type. Let $p_0\left(x\right)\subseteq p\left(x\right)$ be a finite subset of $p\left(x\right)$. We need to find a $b\in\omega$ that satisfies all the formulas in $p_0$. Well, we can just take the successor of the largest ordinal mentioned in the set of formulas $p_0\left(x\right)$. Then this will clearly contain all the ordinals mentioned in $p_0\left(x\right)$. Thus we have that $p\left(x\right)$ is a type. Next, note that $p\left(x\right)$ is not realized in $\mathcal$. For, if it were there would be some $n\in\omega$ that contains every element of $\omega$. If we wanted to realize the type, we might be tempted to consider the model $\langle \omega+1,\in_\rangle$, which is indeed a supermodel of $\mathcal$ that realizes the type. Unfortunately, this extension is not elementary, that is, this model does not have to satisfy $\mathcal$. In particular, the sentence $\exists x \forall y \left(y\in x \lor y=x\right)$ is satisfied by this model and not by $\mathcal$. 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 $\mathcal\text{'}$. The domain of the structure will be $\omega \cup \mathbb\text{'}$ where $\mathbb\text{'}$ is the set of integers adorned in such a way that $\mathbb\text{'}\cap\omega=\emptyset$. Let $<$ denote the usual order of $\mathbb\text{'}$. We interpret the symbol $\in$ in our new structure by $\in_ = \in_ \cup < \cup \,\left(\omega \times \mathbb\text{'}\right)$. The idea being that we are adding a "$\mathbb$-chain", or copy of the integers, above all the finite ordinals. Clearly any element of $\mathbb\text{'}$ realizes the type $p\left(x\right)$. 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 $\,\!x\ne 1+1+1$, $x\le 1+1+1+1+1$, and
square root of 2 The square root of 2 (approximately 1.4142) is a positive real number that, when multiplied by itself, equals the number 2. It may be written in mathematics as \sqrt or 2^, and is an algebraic number. Technically, it should be called the prin ...
are consistent with the axioms 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. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered fiel ...
s, and can be extended to a complete type. This type is not realized in the ordered field of rational numbers, but is realized in the ordered field of reals. Similarly, the infinite set of formulas (over the empty set) is not realized in the ordered field of real numbers, but is realized in the ordered field of
hyperreals In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
. If we allow parameters, for instance all of the reals, we can specify a type $\$ that is realized by an
infinitesimal In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally refer ...
hyperreal that violates the Archimedean property. The reason it is useful to restrict the parameters to a certain subset of the model is that it helps to distinguish the types that can be satisfied from those that cannot. For example, using the entire set of real numbers as parameters one could generate an uncountably infinite set of formulas like $x\ne 1$, $x\ne \pi$, ... that would explicitly rule out every possible real value for ''x'', and therefore could never be realized within the real numbers.

# Stone spaces

It is useful to consider the set of complete ''n''-types over ''A'' as a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called poi ...
. Consider the following equivalence relation on formulas in the free variables ''x''1,…, ''x''''n'' with parameters in ''A'': :$\psi \equiv \phi \Leftrightarrow \mathcal \models \forall x_1,\ldots,x_n \left(\psi\left(x_1,\ldots,x_n\right) \leftrightarrow \phi\left(x_1,\ldots,x_n\right)\right).$ One can show that $\psi \equiv \phi$ if and only if they are contained in exactly the same complete types. The set of formulas in free variables ''x''1,…,''x''''n'' over ''A'' up to this equivalence relation is a
Boolean algebra In mathematics and mathematical logic, Boolean algebra is a branch of algebra. It differs from elementary algebra in two ways. First, the values of the variables are the truth values ''true'' and ''false'', usually denoted 1 and 0, whereas in ...
(and is canonically isomorphic to the set of ''A''-definable subsets of ''M''''n''). The complete ''n''-types correspond to
ultrafilter In the mathematical field of order theory, an ultrafilter on a given partially ordered set (or "poset") P is a certain subset of P, namely a maximal filter on P; that is, a proper filter on P that cannot be enlarged to a bigger proper filter o ...
s of this Boolean algebra. The set of complete ''n''-types can be made into a topological space by taking the sets of types containing a given formula as basic open sets. This constructs the Stone space, which is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Brit ...
, Hausdorff, and totally disconnected. Example. The complete theory 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 . Examples As an example, the field of real numbers is not algebraically closed, because ...
s of characteristic 0 has quantifier elimination, which allows one to show that the possible complete 1-types (over the empty set) correspond to: *
Root In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the sur ...
s of a given irreducible non-constant polynomial over the rationals with leading coefficient 1. For example, the type of square roots of 2. Each of these types is an open point of the Stone space. *Transcendental elements, that are not roots of any non-zero polynomial. This type is a point in the Stone space that is closed but not open. In other words, the 1-types correspond exactly to the prime ideals of the polynomial ring Q 'x''over the rationals Q: if ''r'' is an element of the model of type ''p'', then the ideal corresponding to ''p'' is the set of polynomials with ''r'' as a root (which is only the zero polynomial if ''r'' is transcendental). More generally, the complete ''n''-types correspond to the prime ideals of the polynomial ring Q 'x''1,...,''x''n in other words to the points of the prime spectrum of this ring. (The Stone space topology can in fact be viewed as the
Zariski topology In algebraic geometry and commutative algebra, the Zariski topology is a topology which is primarily defined by its closed sets. It is very different from topologies which are commonly used in the real or complex analysis; in particular, it is n ...
of a
Boolean ring In mathematics, a Boolean ring ''R'' is a ring for which ''x''2 = ''x'' for all ''x'' in ''R'', that is, a ring that consists only of idempotent elements. An example is the ring of integers modulo 2. Every Boolean ring gives rise to a Boolean al ...
induced in a natural way from the Boolean algebra. While the Zariski topology is not in general Hausdorff, it is in the case of Boolean rings.) For example, if ''q''(''x'',''y'') is an irreducible polynomial in two variables, there is a 2-type whose realizations are (informally) pairs (''x'',''y'') of elements with ''q''(''x'',''y'')=0.

# Omitting types theorem

Given a complete ''n''-type ''p'' one can ask if there is a model of the theory that omits ''p'', in other words there is no ''n''-tuple in the model that realizes ''p''. If ''p'' is an
isolated point ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equiva ...
in the Stone space, i.e. if is an open set, it is easy to see that every model realizes ''p'' (at least if the theory is complete). The omitting types theorem says that conversely if ''p'' is not isolated then there is a countable model omitting ''p'' (provided that the language is countable). Example: In the theory of algebraically closed fields of characteristic 0, there is a 1-type represented by elements that are transcendental over the field (mathematics)#Subfields and prime fields, prime field. This is a non-isolated point of the Stone space (in fact, the only non-isolated point). The field of algebraic numbers is a model omitting this type, and the algebraic closure of any transcendental extension of the rationals is a model realizing this type. All the other types are "algebraic numbers" (more precisely, they are the sets of first-order statements satisfied by some given algebraic number), and all such types are realized in all algebraically closed fields of characteristic 0.

# References

* * * {{Mathematical logic Concepts in logic Mathematical logic Model theory