Type (model theory)
In model theory and related areas of mathematics, a type is an object that describes how a element or finite collection of elements in a mathematical structure might behave. More precisely, it is a set of first-order formulas in a language L with free variables x1, x2,…, xn 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 models and omitting types.
Formal definition
Consider a structure for a language L. Let M be the universe of the structure. For every A ⊆ M, let L be the language obtained from L by adding a constant ca for every a ∈ A. In other words,A 1-type over A is a set p of formulas in L with at most one free variable x such that for every finite subset p0 ⊆ p there is some b ∈ M, depending on p0, with .
Similarly an n-type over A is defined to be a set p = p of formulas in L, each having its free variables occurring only among the given n free variables x1,…,xn, such that for every finite subset p0 ⊆ p there are some elements b1,…,bn ∈ M with.
A complete type of over A is one that is maximal with respect to inclusion. Equivalently, for every either or . 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.
An n-type p is said to be realized in if there is an element b ∈ Mn such that. 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, rather than in itself.
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 is said to be isolated by , for, if . Since finite subsets of a type are always realized in, there is always an element b ∈ Mn such that φ is true in ; i.e., 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.
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. Let 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 where is the set of integers adorned in such a way that. Let 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,, and. This is an example of an isolated type, since, working over the theory of the naturals, the formula implies all other formulas that are true about the number 2.
As a further example, the statements
and
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 is not realized in the ordered field of real numbers, but is realized in the ordered field of hyperreals. If we allow parameters, for instance all of the reals, we can specify a type 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,,... 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 formulas in the free variables x1,…, xn with parameters in A:One can show that if and only if they are contained in exactly the same complete types.
The set of formulas in free variables x1,…,xn over A up to this equivalence relation is a Boolean algebra. 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.
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 that realizes p.If p is an isolated point in the Stone space, i.e. if is an open set, it is easy to see that every model realizes p. The omitting types theorem says that conversely if p is not isolated then there is a countable model omitting p.
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. 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", and all such types are realized in all algebraically closed fields of characteristic 0.