Type (model theory)
In the model theory of a type called a lot of first -stage formulas in a language with free variables do not imply a contradiction. Figuratively speaking, specifies a fixed type specific properties that should have an element. Such an element does not necessarily exist, but the properties must not contradict each other, so that at least in a larger structure, such an element can be found. Also expresses a type which elements can not be distinguished by first -stage formulas.
- 4.1 Example
Definition
Be a structure for a language and their universe. For all subsets denote the language obtained from, if one adds a constant for each symbol, ie.
The constant symbols are interpreted by the element, and this expanded structure will be denoted by.
A 1- type (from) over is a set of formulas in at most one free variable (hence 1 type) such that for every finite subset an element exists with, for so all formulas are fulfilled in when looking for the element is used.
Similarly, a type (of ) over a set of formulas, so that for each finite subset of elements exist with.
One type is called complete if it is maximal with respect to inclusion. Therefore applies to any or either For a complete type. A type that is not complete, ie partial type. Partial types can always be added to complete the set of types according to Lindenbaum (or the lemma of Zorn).
Terms
A type is realized in, if there is an element with. After the compactness theorem there exists for every type of an elementary extension in which there is such an implementation. Where a complete type in realized, it is referred to as the complete type over.
One type is isolated by, if the formula for a is consistent with the underlying theory ( ie is a ( partial) type) and the other has the property to imply all formulas in. As a realization possesses, there is an element, so in true, that realizes the entire insulated type. In particular, isolated types have a realization in each elementary substructure or extension.
Examples
Infinitely large natural numbers
Be a model of the natural numbers with the universe and the languages, as the usual order is interpreted. Then is a type over: By definition applies anyway. For a non-empty finite subset we determine the largest natural number that occurs and get into.
The amount is a partial type and says: "is greater than any natural number. " Within the universe of there is no such element, the type is therefore not feasible in. However, there are structures in which it is realized, we can take as the universe, in which a copy of, and to disjoint. So are all the numbers from less than all of the numbers and within the two quantities is considered the usual order.
Is an elementary extension of and. Therefore realized 0 ' type. Also, the type can not be isolated, because otherwise he would already isolated in elementary substructure and thus would have to be implemented there.
We could have just to add another element and can make it the largest element. Also in this structure had an implementation. But in this case would be through isolated. Thus, this extension can not be elementary.
Type of the natural number 2
Hyper Real Numbers
The type is not realized within the real numbers after the Archimedean axiom. An extension of this type is realized, are the hyper real numbers. When this type is realized, the type of infinitely small numbers is realized automatically.
Stone - space
On the set of complete types over can define a topology:
If we denote by the set of all complete types that contain the formula as an element, and they stand for a true or false statement, shall be deemed
In particular, form the basis of a topology. This provides the Stone space. This is compact, Hausdorff and totally disconnected. Isolated types correspond to just the isolated points.
Example
The complete theory of algebraically closed field of characteristic 0 has quantifier elimination, so you can easily determine all types (over the empty set ):
- Types of algebraic numbers: These guys are open points in the Stone space. The insulating formula results from the minimal polynomial. Two algebraic numbers then have exactly the same type if they are conjugate, ie have the same minimal polynomial.
- The type of transcendent elements: This type is not open as a point in the Stone space. All transcendental elements of the same type, this means for every polynomial except for the 0 - polynomial, that the element is not a zero of the polynomial.
Realizations of types in models
While isolated types in each model have a realization, it depends on the other types of model, whether realized or not. It is therefore natural to investigate models in which a particularly large or particularly a few types are realized.
A model that realizes all types over finite sets is called - saturated. Generally, one can define the notion of arbitrary infinite cardinal numbers: A model called - saturated if all types are implemented via small quantities with cardinality. A model is called saturated if it is saturated -.
Conversely makes the Omitting Types Theorem ( in German rarely as type avoidance rate ) agreed, stating that there are models in which a given type has no implementation. It is said that the type is omitted or avoided by the model: Be a non-isolated type in a countable language. Then there is a countable model in which it is not realized. Generally, one can also show that even a countable set of uninsulated types can be omitted.