Type (model theory)

In model theory and related areas of mathematics, a type is a set of first-order formulas in a language L with free variables x₁, x₂,…, x_n which are true of a sequence of elements of an L-structure $\mathcal{M}$. Loosely speaking, types describe possible elements of a mathematical structure. Depending on the context, types can be complete or partial and they may use a fixed set of constants, A, from the structure $\mathcal{M}$. The question of which types represent actual elements of $\mathcal{M}$ leads to the ideas of saturated models and omitting types.

Definition

Consider a structure $\mathcal{M}$ for a language L. Let M be the universe of the structure. For every A ⊆ M, let L(A) be the language which is obtained from L by adding a constant c_a for every a ∈ A. In other words,

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

A 1-type (of $\mathcal{M}$) 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 p₀(x) ⊆ p(x) there is some b ∈ M, depending on p₀(x), with $\mathcal{M} \models p_0(b)$ (i.e. all formulas in p₀(x) are true in $\mathcal{M}$ when x is replaced by b).

Similarly an n-type (of $\mathcal{M}$) over A is defined to be a set p(x₁,…,x_n) = p(x) of formulas in L(A) such that for every finite subset p₀(x) ⊆ p(x) there are some elements b₁,…,b_n ∈ M with $\mathcal{M}\models p_0(b_1,\ldots,b_n)$.

Complete type refers to those types which are maximal with respect to inclusion, i.e. if p(x) is a complete type, then $\forall \phi(\boldsymbol{x}) \in L(A,\boldsymbol{x})$ either $\phi(\boldsymbol{x}) \in p(\boldsymbol{x})$ or $\lnot\phi(\boldsymbol{x}) \in p(\boldsymbol{x})$. 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 $\mathcal{M}$ if there is an element b ∈ Mⁿ such that $\mathcal{M}\models p(\boldsymbol{b})$. 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 $\mathcal{M}$, rather than in $\mathcal{M}$ itself. If a complete type is realized by b in $\mathcal{M}$, then the type is typically denoted $tp_n^{\mathcal{M}}(\boldsymbol{b}/A)$ and referred to as the complete type of b over A.

A type p(x) is said to be isolated by φ if there is a formula φ(x) with the property that $\forall \psi(\boldsymbol{x}) \in p(\boldsymbol{x}), \varphi(\boldsymbol{x}) \rightarrow \psi(\boldsymbol{x})$. Since finite subsets of a type are always realized in $\mathcal{M}$, there is always an element b ∈ Mⁿ such that φ(b) is true in $\mathcal{M}$; i.e. $\mathcal{M} \models \varphi(\boldsymbol{b})$, 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

Let $\mathcal{N}$ have the natural numbers $\mathbb{N}$ as domain which are ordered by $\leq$. Then $p(x):=\{ n \leq x \mid n\in \mathbb{N}\}$ is a type of $\mathcal{N}$ over $\mathbb{N}$: By definition $\mathcal{N}\models \emptyset$ anyhow. For every non-empty finite subset $p_0(x)\subseteq p(x)$, we can determine the maximal natural number $m_{p_0}$ occurring in p₀ and obtain $\mathcal{N}\models p_0(m_{p_0})$.

The set p(x) is a partial type and claims "x is greater than any natural number". Yet such an element does not exist in the domain of $\mathcal{N}$. This can be remedied by a new structure, $\mathcal{N}^*$ say, with domain $\mathbb{N}\cup\mathbb{N}'$ where $\mathbb{N}\cap \mathbb{N}'=\emptyset$ and $\leq^*:=\leq \cup \leq'\cup\, \mathbb{N}\times \mathbb{N}'$.

$\mathcal{N}^*$ is an elementary extension of $\mathcal{N}$ (w.r.t. $L(\mathbb N )$) and $\mathcal{N}^*\models p(0')$. Hence 0' realises p(x). Of course, p(x) cannot be isolated in $\mathcal{N}^*$ or else it would be isolated in $\mathcal{N}$ since this is an elementary substructure of $\mathcal{N}^*$. But being isolated in $\mathcal{N}$ means being realised in $\mathcal{N}$.

Alternatively one could simply add another element $c\notin \mathbb{N}$ to $\mathbb{N}$ and make c the greatest element. But then p(x) would be isolated by $\exists x. \forall y. y\leq x$ in the extension of $\mathcal N$. Hence this extension cannot be an elementary extension of $\mathcal N$.

Another example: the complete type of the number 2 over the emptyset, considered as a member of the natural numbers, would be the set of all first-order statements describing a variable x that are true for x = 2. This set would include formulas such as $\,\!x\ne 1+1+1$, $x\le 1+1+1+1+1$, and $\exists y(y<x)$. This is an example of an isolated type, since the formula x = 1 + 1 implies all other formulas that are true about the number 2.

For example, the statements

$\forall y(y^2<2 \implies y<x)$

and

$\forall y((y>0 \and y^2>2) \implies y>x)$

describing the square root of 2 are consistent with the axioms of ordered 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 emptyset) {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 of hyperreals. If we allow more parameters, for instance all of the reals, we can specify a type $\{ 0 < x < r : r \in \mathbb{R} \}$ that is realized by an infinitesimal 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. Consider the following equivalence relation on formulae in the free variables x₁,…, x_n with parameters in M:

$\psi \equiv \phi \Leftrightarrow \mathcal{M} \models \forall x_1,\ldots,x_n (\psi(x_1,\ldots,x_n) \leftrightarrow \phi(x_1,\ldots,x_n)).$

One can show that $\psi \equiv \phi$ iff they are contained in exactly the same complete types.

The set of formulae in free variables x₁,…,x_n over A up to this equivalence relation is a Boolean algebra (and is canonically isomorphic to the set of A-definable subsets of Mⁿ). The complete n-types correspond to ultrafilters 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, Hausdorff, and totally disconnected.

Example. The complete theory of algebraically closed fields of characteristic 0 has quantifier elimination which allows one to show that the possible complete 1-types correspond to:

• Roots 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. More generally, the complete n-types correspond to the prime ideals of the polynomial ring Q[x₁,...,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 of a Boolean ring induced in a natural way from the lattice structure of 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 2 variables, there is a 2-type whose realizations are (informally) pairs (x,y) of transcendental elements with q(x,y)=0.

The 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 which realizes p. If p is an isolated point in the Stone space, i.e. if {p} 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 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

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