Model (logic)

In the mathematical logic one is model of a axiom system a provided with certain structures amount to which the axioms of this system apply.

For example, the geometry of the Euclidean plane is a model of the Euclidean axiom system, and if the axiom of parallels is dispensed with, the geometry of the hyperbolic plane is also a model of the remaining axiom system.

The existence of a model proves the consistency of a system of axioms. The model theory is concerned with what models there are certain axioms.

Definition of models

In an elementary language , the expressions or formulas are formed using an alphabet consisting of the following basic characters. ${\displaystyle L}$ ${\displaystyle L}$

1. countably many individual variables: ${\displaystyle x_{1},x_{2},x_{3},\ldots }$
2. Function sign: ${\displaystyle f_{1},f_{2},f_{3},\ldots }$
3. Relation sign: ${\displaystyle R_{1},R_{2},R_{3},\ldots }$
4. Individual sign: ${\displaystyle c_{1},c_{2},c_{3},\ldots }$
5. Logical signs: ¬, ∧, ∨, →, ↔, ∃, ∀, =
6. technical characters: (,)

If an algebraic structure, then an elementary language suitable for A contains a function sign for every function , a relational sign for every relation, and an individual sign for every element . The signature of the algebraic structure consists of the families of all digits of the function or relation signs and that of all individual signs. If it matches the signature of an elementary language , then this is suitable for formulating statements about the algebraic structure. If corresponding functions, relations or elements are assigned to the function, relation and individual signs, then the language in the structure is interpreted. ${\displaystyle (A,F^{A},R^{A},C^{A})}$ ${\displaystyle f_{i}^{A}\in F^{A}}$ ${\displaystyle f_{i}}$ ${\displaystyle R_{i}^{A}\in R^{A}}$ ${\displaystyle R_{i}}$ ${\displaystyle c_{i}^{A}\in C^{A}}$ ${\displaystyle c_{i}}$ ${\displaystyle L}$ ${\displaystyle C^{A}}$

Terms are defined inductively in that individual variables and individual signs are terms and there is also a term for terms and one -place function sign. Expressions are also defined inductively: ${\displaystyle t_{1},\ldots ,t_{n}}$ ${\displaystyle n}$ ${\displaystyle f}$ ${\displaystyle f(t_{1},\ldots ,t_{n})}$

- for terms and one -place relational sign is an expression ${\displaystyle t_{1},\ldots ,t_{n}}$ ${\displaystyle n}$ ${\displaystyle R}$ ${\displaystyle R(t_{1},\ldots ,t_{n})}$

- Term equations are expressions ${\displaystyle t_{1}=t_{2}}$

- if φ and ψ are expressions, then ¬φ, φ ∧ ψ, φ ∨ ψ, φ → ψ, φ ↔ ψ are also expressions

- if φ is an expression in which the character series ∃x or ∀x does not occur, then ∃xφ and ∀xφ are also expressions.

Statements are expressions in which there are no free variables. The free occurrence of a variable is again defined inductively: the individual variable occurs free in the expression φ if and only if ${\displaystyle x}$

- φ is atomic and x occurs in φ, or

- φ has the form ¬ψ and x occurs free in ψ, or

- φ has the form ψ ∧ χ, ψ ∨ χ, ψ → χ or ψ ↔ χ and x occurs freely in ψ or χ, or

- φ has the form ∃yψ or ∀yψ, and x occurs free in ψ and x, y are different individual variables.

A ⊨ φ denotes the validity of φ in structure A. This is again defined inductively:

- if φ is an atomic statement, then ? ⊨ φ is already defined by the interpretation

- ? ⊨ ¬φ ⇔ φ does not hold in ?, ? ⊨ φ ∧ ψ ⇔ ? ⊨ φ and ? ⊨ ψ, ? ⊨ φ ∨ ψ ⇔ ? ⊨ φ or ? ⊨ ψ, ? ⊨ φ → ψ ⇔ if ? ⊨ φ , so ? ⊨ ψ, ? ⊨ φ ↔ ψ ⇔ ? ⊨ φ if and only if ? ⊨ ψ

- ? ⊨ ∃xφ (x) ⇔ there is an element a in ? such that ? ⊨ φ (a), ? ⊨ ∀ xφ (x) ⇔ for all elements a in ? ? ⊨ φ (a)

An expression is valid if applies to all elements , that is, if the statement applies. A set T of expressions or statements from L that is deductively closed is called an elementary theory . ${\displaystyle \phi (x_{1},\ldots ,x_{n})}$ ${\displaystyle A}$ ${\displaystyle A\models (a_{1},\ldots ,a_{n})}$ ${\displaystyle a_{1},\ldots ,a_{n}\in {\mathcal {A}}}$ ${\displaystyle \forall x_{1}\ldots \forall x_{n}\phi (x_{1},\ldots ,x_{n})\in {\mathcal {A}}}$

If T is a theory formulated in L and ? is a structure for L, that is, ? and L have the same signature, then ? is a model of T if all statements or expressions from T are valid in ? (in symbols ? ⊨ T) .

literature

• CC Chang, HJ Keisler: "Model Theory." Studies in Logic and the Foundations of Mathematics 73, North-Holland Amsterdam (1973).

Web links

• Lexicon of Mathematics: Model Theory , Springer Verlag (2017).