Elementary class

The term elementary class belongs to the model theory, a branch of mathematical logic. It is about the question of how to characterize classes of structures by sets of first-order predicate logic.

Definitions

Is a language of first-order logic and is a set of language, it should be the class of all structures that satisfy the sentence, that is, for which it holds (for the Herleitbarkeitsbegriff see Article first-order logic ). One says in this case, is a model for. A class of S- structures is called elementary if there is a set, so that it coincides with. The members of the Class leave in the first order predicate logic characterized by the set

Often a single sentence is not enough to characterize a class of structures. Was for a non-empty set of sentences from

The class of all S- structures which meet all the records from. This is called a class - elementary if there is a non-empty set of sentences so that they. Coincides with, said to commemorate above average education Is finite, then there is an elementary class, as is apparently

Examples and phrases

A typical example of an elementary class is the class of all body. As a symbol set is used and as you just take the conjunction of all the field axioms.

To specify an example of a - elementary class, we consider again the symbol set, the conjunction of all the field axioms and for every prime number with the designated set, with many ones are added on the left side. The sentence apparently characterizes the elementary class of fields of characteristic. The infinite set

Then defined the class of all field of characteristic 0, that is, elementary therefore. One can show that this class is not elementary.

Finally, there are important classes that are not even - elemental, so for example the class of all finite field. The reason for this is the following sentence:

• Contains an elementary - Class S- structures of arbitrarily large finite cardinality, so it also contains infinite S- structures.

A - elementary class that includes all finite body containing the residue class bodies such arbitrarily large finite cardinality, and hence after this block also infinite, which therefore do not belong to the class under consideration.

In addition:

• Contains a - elementary class an infinite S- structure, it also contains S- structures of arbitrarily large cardinality.

In particular, elementary - contained classes in the situation of the last sentence of non- isomorphic structures, because isomorphic structures necessarily have the same cardinality. Therefore, it can not succeed, the set of natural numbers, or the parent field of real numbers, who are both up to isomorphism uniquely to characterize by a set of sentences of first-order predicate logic. This finding further then leads to non-standard models and non-standard analysis.

Axiomatization

It is said that one - elementary class that is given by a set of propositions is, axiomatized by, and the individual records in the names of the axioms of the class. This is a synonym for - elementary axiomatizable. Some authors do not distinguish between elementary and elementary but generally speak of axiomatization. The above defined elementarity then corresponds to a finite axiomatization.