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Inmodel theory and related areas ofmathematics, atype is an object that describes how a (real or possible) element or finite collection of elements in amathematical structure might behave. More precisely, it is a set offirst-order formulas in a languageL withfree variablesx₁,x₂,...,x_n that are true of a set ofn-tuples of anL-structure${\displaystyle \mathcal{M}}$. Depending on the context, types can becomplete orpartial and they may use a fixed set of constants,A, from the structure${\displaystyle \mathcal{M}}$. The question of which types represent actual elements of${\displaystyle \mathcal{M}}$ leads to the ideas ofsaturated models andomitting types.

Consider astructure${\displaystyle \mathcal{M}}$ for alanguageL. LetM be theuniverse of the structure. For everyA ⊆ M, letL(A) be the language obtained fromL by adding a constantc_a for everya ∈ A. In other words,

${\displaystyle L(A)=L\cup \{c_a:a\in A\}.}$

A1-type (of${\displaystyle \mathcal{M}}$) overA is a setp(x) of formulas inL(A) with at most one free variablex (therefore 1-type) such that for every finite subsetp₀(x) ⊆ p(x) there is someb ∈ M, depending onp₀(x), with${\displaystyle \mathcal{M}\models p_0(b)}$ (i.e. all formulas inp₀(x) are true in${\displaystyle \mathcal{M}}$ whenx is replaced byb).

Similarly ann-type (of${\displaystyle \mathcal{M}}$) overA is defined to be a setp(x₁,...,x_n) = p(x) of formulas inL(A), each having its free variables occurring only among the givenn free variablesx₁,...,x_n, such that for every finite subsetp₀(x) ⊆ p(x) there are some elementsb₁,...,b_n ∈ M with${\displaystyle \mathcal{M}\models p_0(b_1,\dots ,b_n)}$.

Acomplete type of${\displaystyle \mathcal{M}}$ overA is one that ismaximal with respect toinclusion. Equivalently, for every${\displaystyle \varphi (\mathit{x})\in L(A,\mathit{x})}$ either${\displaystyle \varphi (\mathit{x})\in p(\mathit{x})}$ or${\displaystyle \mathrm{\neg }\varphi (\mathit{x})\in p(\mathit{x})}$. Any non-complete type is called apartial type. So, the wordtype in general refers to anyn-type, partial or complete, over any chosen set of parameters (possibly the empty set).

Ann-typep(x) is said to berealized in${\displaystyle \mathcal{M}}$ if there is an elementb ∈ Mⁿ such that${\displaystyle \mathcal{M}\models p(\mathit{b})}$. The existence of such a realization is guaranteed for any type by thecompactness theorem, although the realization might take place in someelementary extension of${\displaystyle \mathcal{M}}$, rather than in${\displaystyle \mathcal{M}}$ itself. If a complete type is realized byb in${\displaystyle \mathcal{M}}$, then the type is typically denoted${\displaystyle t{p}_n^{\mathcal{M}}(\mathit{b}/A)}$ and referred to asthe complete type ofb overA.

A typep(x) is said to beisolated by${\displaystyle \phi }$, for${\displaystyle \phi \in p(x)}$, if for all${\displaystyle \psi (\mathit{x})\in p(\mathit{x}),}$ we have${\displaystyle \mathrm{Th}(\mathcal{M})\models \phi (\mathit{x})\to \psi (\mathit{x})}$. Since finite subsets of a type are always realized in${\displaystyle \mathcal{M}}$, there is always an elementb ∈ Mⁿ such thatφ(b) is true in${\displaystyle \mathcal{M}}$; i.e.${\displaystyle \mathcal{M}\models \phi (\mathit{b})}$, thusb 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 asaturated model, and theultrapower construction provides one way of producing saturated models.

Consider the languageL with one binaryrelation symbol, which we denote as${\displaystyle \in }$. Let${\displaystyle \mathcal{M}}$ be the structure${\displaystyle ⟨\omega ,{\in }_{\omega }⟩}$ for this language, which is theordinal${\displaystyle \omega }$ with its standardwell-ordering. Let${\displaystyle \mathcal{T}}$ denote thefirst-order theory of${\displaystyle \mathcal{M}}$.

Consider the set ofL(ω)-formulas${\displaystyle p(x):=\{n{\in }_{\omega }x\mid n\in \omega \}}$. First, we claim this is a type. Let${\displaystyle p_0(x)\subseteq p(x)}$ be a finite subset of${\displaystyle p(x)}$. We need to find a${\displaystyle b\in \omega }$ that satisfies all the formulas in${\displaystyle p_0}$. Well, we can just take the successor of the largest ordinal mentioned in the set of formulas${\displaystyle p_0(x)}$. Then this will clearly contain all the ordinals mentioned in${\displaystyle p_0(x)}$. Thus we have that${\displaystyle p(x)}$ is a type. Next, note that${\displaystyle p(x)}$ is not realized in${\displaystyle \mathcal{M}}$. For, if it were there would be some${\displaystyle n\in \omega }$ that contains every element of${\displaystyle \omega }$. If we wanted to realize the type, we might be tempted to consider the structure${\displaystyle ⟨\omega +1,{\in }_{\omega +1}⟩}$, which is indeed an extension of${\displaystyle \mathcal{M}}$ that realizes the type. Unfortunately, this extension is not elementary, for example, it does not satisfy${\displaystyle \mathcal{T}}$. In particular, the sentence${\displaystyle \mathrm{\exists }x\mathrm{\forall }y(y\in x\vee y=x)}$ is satisfied by this structure and not by${\displaystyle \mathcal{M}}$.

So, we wish to realize the type in an elementary extension. We can do this by defining a newL-structure, which we will denote${\displaystyle {\mathcal{M}}^{\prime }}$. The domain of the structure will be${\displaystyle \omega \cup {\mathbb{Z}}^{\prime }}$ where${\displaystyle {\mathbb{Z}}^{\prime }}$ is the set of integers adorned in such a way that${\displaystyle {\mathbb{Z}}^{\prime }\cap \omega =\mathrm{\varnothing }}$. Let denote the usual order of${\displaystyle {\mathbb{Z}}^{\prime }}$. We interpret the symbol${\displaystyle \in }$ in our new structure by. The idea being that we are adding a "${\displaystyle \mathbb{Z}}$-chain", or copy of the integers, above all the finite ordinals. Clearly any element of${\displaystyle {\mathbb{Z}}^{\prime }}$ realizes the type${\displaystyle p(x)}$. 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 (in the language ofPeano arithmetic), describing a variablex, that are true whenx = 2. This set would include formulas such as${\displaystyle x\ne 1+1+1}$,${\displaystyle x\le 1+1+1+1+1}$, and. This is an example of an isolated type, since, working over the theory of the naturals, the formula${\displaystyle x=1+1}$ implies all other formulas that are true about the number 2.

As a further example, the statements

and

${\displaystyle \mathrm{\forall }y((y>0\wedge y^2>2)⟹y>x)}$

describing thesquare root of 2 are consistent with the axioms ofordered fields, 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) {x>1, x>1+1, x>1+1+1, ...} is not realized in the ordered field of real numbers, but is realized in the ordered field ofhyperreals. Similarly, we can specify a type that is realized by aninfinitesimal hyperreal that violates theArchimedean 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${\displaystyle x\ne 1}$,${\displaystyle x\ne \pi }$, ... that would explicitly rule out every possible real value forx, and therefore could never be realized within the real numbers.

It is useful to consider the set of completen-types overA as atopological space. Consider the followingequivalence relation on formulas in the free variablesx₁,...,x_n with parameters inA:

${\displaystyle \psi \equiv \varphi \iff \mathcal{M}\models \mathrm{\forall }{x}_1,\dots ,x_n(\psi (x_1,\dots ,x_n)\leftrightarrow \varphi (x_1,\dots ,x_n)).}$

One can show that${\displaystyle \psi \equiv \varphi }$ if and only if they are contained in exactly the same complete types.

The set of formulas in free variablesx₁,...,x_n overA up to this equivalence relation is aBoolean algebra (and is canonicallyisomorphic to the set ofA-definable subsets ofMⁿ). The completen-types correspond toultrafilters of this Boolean algebra. The set of completen-types can be made into a topological space by taking the sets of types containing a given formula as abasis of open sets. This constructs theStone space associated to the Boolean algebra, which is acompact,Hausdorff, andtotally disconnected space.

Example. Thecomplete theory ofalgebraically closed fields ofcharacteristic 0 hasquantifier elimination, which allows one to show that the possible complete 1-types (over the empty set) correspond to:

• Roots of a givenirreducible non-constant polynomial over the rationals with leading coefficient 1. For example, the type of square roots of 2. Each of these types is anisolated point of the Stone space.
• Transcendental elements, which are not roots of any non-zero polynomial. This type is a point in the Stone space that is closed but not isolated.

In other words, the 1-types correspond exactly to theprime ideals of thepolynomial ringQ[x] over the rationalsQ: ifr is an element of the model of typep, then the ideal corresponding top is the set of polynomials withr as a root (which is only the zero polynomial ifr is transcendental). More generally, the completen-types correspond to the prime ideals of the polynomial ringQ[x₁,...,x_n], in other words to the points of theprime spectrum of this ring. (The Stone space topology can in fact be viewed as theZariski topology of aBoolean ring 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, ifq(x,y) is an irreducible polynomial in two variables, there is a 2-type whose realizations are (informally) pairs (x,y) of elements withq(x,y)=0.

Given a completen-typep one can ask if there is a model of the theory thatomitsp, in other words there is non-tuple in the model that realizesp. Ifp is anisolated point in the Stone space, i.e. if {p} is an open set, it is easy to see that every model realizesp (at least if the theory is complete). Theomitting types theorem says that conversely ifp is not isolated then there is a countable model omittingp (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 theprime fieldQ. 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 thealgebraic closure of anytranscendental 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.

• Hodges, Wilfrid (1997).A shorter model theory.Cambridge University Press.ISBN 0-521-58713-1.
• Chang, C.C.;Keisler, H. Jerome (1989).Model Theory (3rd ed.).Elsevier.ISBN 0-7204-0692-7.
• Marker, David (2002).Model Theory: An Introduction.Graduate Texts in Mathematics. Vol. 217. Springer.ISBN 0-387-98760-6.

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