Type (model theory) Totally Explained

Everything about Type Model Theory totally explained

In model theory, consider a structure of signature σ,

M

, a set of parameters

A subset M

and some tuple of elements of

M

ar isn't realized in the real numbers, but is realized in the hyperreals . 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.
   A

partial $n$ -type over

A

is any consistent (but not necessarily maximal) set of formulae with parameters in

A

and free variables

x_1,ldots,x_n

. Any complete type is a partial type. Conversely, it can be proved that, any partial type is a subset of a complete type. The word type is sometimes used to mean complete type and sometimes any partial type.
The reason it's useful to restrict the constants 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 constants one could generate an uncountably infinite list of formulae like

x
e 1

x
e 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

x_1,ldots,x_n

with parameters in

M

psi equiv phi

iff

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're contained in exactly the same complete types.
The set of formulae in free variables

x_1,ldots,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

M^n

). 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 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's 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 corrresponding 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,...,x₁], in other words to the points of the prime spectrum of this ring. (But the Stone space topology isn't the Zariski topology, which isn't Hausdorff.) For example, if q(x,y) is an irreducible polynomial in 2 variables, there's a 2-type whose elements 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's a model of the theory that omits p, in other words there's no n-tuple of type p. If p is isolated in the Stone space, in other words if it's an open point, it's easy to see that every model realizes p (at least if the theory is complete). The omitting types theorem says that conversely if p isn't isolated then there's a countable model omitting p (provided that the language is countable).
Example: In the theory of algebraically closed fields of characteristic 0, there's 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're 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.

Further Information

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