# Axiom system

An axiom system (also: axiomatic system ) is a system of fundamental statements , axioms , which are accepted without proof and from which all theorems (theorems) of the system are logically derived. The derivation takes place through the rules of a formal logical calculus . A theory consists of a system of axioms and all of its theorems derived from it. Mathematical theories are usually axiomatized as elementary language (also: language of the first level with a set of symbols) within the framework of the first level predicate logic .

## General

An axiom system as the product of the axiomatization of a field of knowledge serves the precise, economical and clear "representation of the propositions applicable in it and the inferential relationships between them." At the same time, the axiomatization forces a clear conceptualization. Elements of an axiomatic system are:

1. an alphabet from which the expressions are made according to certain rules;
2. a lot of basic expressions - the axioms - and
3. a system of logical inference rules (calculus) for the derivation of further expressions, the theorems .

## An example: group theory

The group theory is formulated as an elementary language within the framework of the first-level predicate logic.

1. The alphabet: All expressions of the elementary language which - in addition to the logical symbols and equality (shown here with ) - contains the set of symbols ; there is a two-digit function symbol (linkage of group elements) and a constant (single element).${\ displaystyle L ^ {S}}$${\ displaystyle \ equiv}$${\ displaystyle S_ {Gr} = \ {\ circ, e \}}$${\ displaystyle \ circ}$${\ displaystyle e}$
2. The group axioms are
1. ${\ displaystyle \ forall x \ forall y \ forall z ((x \ circ y) \ circ z \ equiv x \ circ (y \ circ z))}$
2. ${\ displaystyle \ forall x (x \ circ e \ equiv x)}$
3. ${\ displaystyle \ forall x \ exists y (x \ circ y \ equiv e)}$
3. The logical system used: The sequential calculus of predicate logic 1st level

The entire group theory can be built on this system of axioms; d. That is, all theorems of group theory can be derived from this.

## Properties of systems of axioms

In the following we denote, as usual, the derivability relation of the underlying logical calculus (sequence calculus, calculus of natural inference) with ; be the corresponding inference operation , which therefore assigns the corresponding theory to every set M of axioms . ${\ displaystyle \ vdash}$${\ displaystyle Cn (\ Gamma) = \ {\ mathrm {A} \ mid \ Gamma \ vdash \ mathrm {A} \}}$ ${\ displaystyle T = Cn (M)}$

The inference operation is a hull operator ; That is, it holds in particular ( idempotency of the envelope operator ). ${\ displaystyle Cn (T) = Cn (Cn (M)) = Cn (M)}$

That is why theories are deductively closed , so one cannot derive anything further from T that cannot already be proven from M. M is also called an axiomatization of T.

### consistency

A set M of axioms (and also the corresponding theory T) is called consistent (or inconsistent ) if one cannot derive any contradictions from these axioms. That means: It is not possible to derive both a proposition A and its negation ¬ A from M (or T) using the rules of the axiom system.

In words of Tarski :

"A deductive discipline is called consistent if no two theorems of this discipline contradict each other or, in other words, if at least one of any two contradicting sentences (...) cannot be proven."

- Tarski

### independence

An expression A is said to be independent of a set M of axioms if A cannot be derived from the axioms in M. Similarly, a set M is independent of axioms if each of the axioms in M ​​is independent of the rest of the axioms:

${\ displaystyle M \ setminus A \ nvdash A}$for everyone . ${\ displaystyle A \ in M}$

In a nutshell: "The axioms are independent if none of them can be derived from the others".

### completeness

A set M of axioms is called complete (also faithful to negation ) if for every sentence A of the language it holds that the sentence A itself or its negation ¬ A can be derived from the axioms in M. It is equivalent to the fact that every extension of M by a previously unprovable theorem becomes contradictory. The same applies to a theory. Complete theories are therefore characterized by the fact that they have no consistent extensions.

Caution: the completeness of a theory or a set of axioms is a purely syntactic property and must not be confused with semantic completeness.

## Models and proofs of consistency, independence and completeness

For the following we assume that the underlying calculus is correct ; d. This means that every syntactic derivation also implies the semantic inference (this is a minimum requirement for an axiomatic system, which e.g. applies to the sequential calculus of first-order predicate logic).

If it has a model for a system of axioms , then M is free of contradictions. Because suppose there was an expression A with and . Every model of M would then be both a model of and of - what cannot be. ${\ displaystyle M \ vdash A}$${\ displaystyle M \ vdash \ neg A}$${\ displaystyle A}$${\ displaystyle \ neg A}$

The consistency thus an axiom system can be shown by specifying a single model. So it follows e.g. B. the consistency of the above axioms of group theory through the specification of the concrete set with and the definition of through the addition modulo 2 ( ). ${\ displaystyle \ {0.1 \}}$${\ displaystyle e = 0}$${\ displaystyle \ circ}$${\ displaystyle 0 \ circ 0 = 0.0 \ circ 1 = 1 \ circ 0 = 1.1 \ circ 1 = 0}$

Models can also be used to show the independence of the axioms of a system: one constructs two models for the subsystem from which a particular axiom A has been removed - one in which A holds and another in which A does not hold.

Two models are called isomorphic if there is a one-to-one correspondence between their elements that contains both relations and functions. A system of axioms for which all models are isomorphic to one another is called categorical . A categorical system of axioms is complete . Because the axiom system is not complete; d. That is, there is an expression A for which neither A nor can be derived from the system. Then there is both a model for and a model for . These two models, which of course are also models for , are not isomorphic. ${\ displaystyle \ neg A}$${\ displaystyle M \ cup \ {A \}}$${\ displaystyle M \ cup \ {\ neg A \}}$${\ displaystyle M}$

## Axiom systems in individual areas

### logic

For elementary propositional logic, first-order predicate logic and various modal logics, there are axiomatic systems that meet the requirements mentioned.

For the predicate logics of higher levels, only consistent, but not complete axiomatic systems can be developed. The decision problem cannot be solved in them.

### arithmetic

For arithmetic, Gödel's incompleteness theorem applies . This is discussed further below.

### geometry

David Hilbert succeeded in axiomatizing Euclidean geometry in 1899 .

### physics

Günther Ludwig laid in the 1980s, an axiomatization of quantum mechanics before

### Linguistics

In 1933, Karl Bühler tried to develop an axiomatic system for linguistics .

### Economic theory

In 1991 Arnis Vilks proposed a system of axioms for neoclassical economic theory .

## Axiomatic system and Gödel's incompleteness theorem

Gödel's incompleteness theorems of 1931 speak of at most recursively enumerable axiomatized theories that are formulated in first-order logic. A complete and correct calculus of evidence is assumed. The first sentence says: If the axioms of arithmetic are consistent, then arithmetic is incomplete. So there is at least one proposition , so that neither its negation nor its negation ¬ can be proven in arithmetic . Furthermore, it can be shown that every extension of the axioms that remains recursively enumerable is also incomplete. Thus the incompleteness of arithmetic is a systematic phenomenon and cannot be remedied by simply expanding the axioms. The second theorem of incompleteness says that especially the consistency of arithmetic cannot be proven in the axiomatic system of arithmetic. ${\ displaystyle \ Phi _ {G}}$${\ displaystyle \ Phi _ {G}}$${\ displaystyle \ Phi _ {G}}$