Elementary class

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

Definitions

If there is a language of logic of the first level and if there is a sentence of this language, then the class of all is - structures that satisfy the sentence , that is, for which applies (for the concept of derivation see article first-level predicate logic ). In this case it is said to be a model for . A class of S-structures is called elementary if there is a sentence such that it coincides with . The members of the class can thus be characterized in the first order predicate logic by the sentence . ${\displaystyle L_{I}^{S}}$ ${\displaystyle \varphi }$ ${\displaystyle \mathrm {Mod} ^{S}\varphi }$ ${\displaystyle S}$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \varphi }$ ${\displaystyle {\mathcal {A}}\vDash \varphi }$ ${\displaystyle \vDash }$ ${\displaystyle {\mathcal {A}}}$ ${\displaystyle \varphi }$ ${\displaystyle \varphi }$ ${\displaystyle \mathrm {Mod} ^{S}\varphi }$ ${\displaystyle \varphi }$

Often a single sentence is not enough to characterize a class of structures. For a non-empty set of sentences from is ${\displaystyle \Phi }$ ${\displaystyle L_{I}^{S}}$

${\displaystyle \mathrm {Mod} ^{S}\Phi \quad :=\quad \bigcap _{\varphi \in \Phi }\mathrm {Mod} ^{S}\varphi }$

the class of all S-structures that satisfy all theorems from . A class is called -elementary if there is a non-empty set of sentences such that it coincides with , which is to remind of the above averaging. Is finite, then there is an elementary class, because is obvious ${\displaystyle \Phi }$ ${\displaystyle \Delta }$ ${\displaystyle \Phi }$ ${\displaystyle \mathrm {Mod} ^{S}\Phi }$ ${\displaystyle \Delta }$ ${\displaystyle \Phi =\{\varphi _{1},\ldots ,\varphi _{n}\}}$

${\displaystyle \mathrm {Mod} ^{S}\{\varphi _{1},\ldots ,\varphi _{n}\}=\mathrm {Mod} ^{S}\varphi _{1}\land \ldots \land \varphi _{n}}$.

Examples and sentences

A typical example of an elementary class is the class of all bodies . As a set of symbols one uses and as one simply takes the conjunction of all body axioms. ${\displaystyle S=\{0,1,+,\cdot \}}$ ${\displaystyle \varphi }$

An example of a specified -elementare class, we again consider the set of symbols , the conjunction of all field axioms and for each prime number to to designated set , where on the left many ones are added. The proposition evidently characterizes the elementary class of bodies of characteristic . The infinite amount ${\displaystyle \Delta }$ ${\displaystyle S=\{0,1,+,\cdot \}}$ ${\displaystyle \varphi _{K}}$ ${\displaystyle p}$ ${\displaystyle \varphi _{p}}$ ${\displaystyle 1+\ldots +1\equiv 0}$ ${\displaystyle p}$ ${\displaystyle \varphi _{K}\land \varphi _{p}}$ ${\displaystyle p}$

${\displaystyle \Phi =\{\varphi _{K}\}\cup \{\neg \varphi _{p};p{\mbox{ Primzahl}}\}}$

then defines the class of all bodies with characteristic 0, which is therefore -elementary. One can show that this class is not elementary. ${\displaystyle \Delta }$

Finally, there are important classes that are not even -elementary, such as the class of all finite fields. The reason for this is the following sentence: ${\displaystyle \Delta }$

• If an elementary class contains S-structures of arbitrarily large finite thickness , it also contains infinite S-structures. ${\displaystyle \Delta }$

An elementary class, which includes all finite fields, contains with the remainder class fields such arbitrarily large finite thicknesses, and thus also infinite ones according to this theorem, which therefore do not belong to the considered class. ${\displaystyle \Delta }$ ${\displaystyle \mathbb {Z} /(p)}$

The following also applies:

• If an elementary class contains an infinite S-structure, it also contains S-structures of arbitrarily large thickness. ${\displaystyle \Delta }$

In particular, elementary classes in the situation of the last sentence contain non-isomorphic structures, because isomorphic structures necessarily have the same cardinality. Therefore it cannot be possible to characterize the set of natural numbers or the ordered field of real numbers, both of which are unique except for isomorphism, by a set of theorems of first-order predicate logic. This finding then leads to non-standard models and non-standard analysis . ${\displaystyle \Delta }$

Axiomatizability

One says that an elementary class given by a set of propositions is axiomatized by, and the individual sentences in are called the axioms of the class. Thus -elementary is synonymous with axiomatizable. Some authors do not differentiate between elementary and elementary , but speak generally of axiomatizability. The elementarity defined above then corresponds to a finite axiomatizability. ${\displaystyle \Delta }$ ${\displaystyle \Phi }$ ${\displaystyle \Phi }$ ${\displaystyle \Phi }$ ${\displaystyle \Delta }$ ${\displaystyle \Delta }$

Individual evidence

1. ^ Heinz-Dieter Ebbinghaus, Jörg Flum, Wolfgang Thomas: Introduction to mathematical logic. Spektrum Akademischer Verlag, Heidelberg / Berlin / Oxford 1996, ISBN 3-8274-0130-5 , in particular Chapter VI, §3
2. ↑ Philipp Rothmaler: Introduction to Model Theory , Spektrum Akademischer Verlag 1995, ISBN 978-3-86025-461-5 , Chapter 3.4