# Axiomatic system

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In mathematics and logic, an **axiomatic system** is any set of primitive notions and axioms to logically derive theorems. A theory is a consistent, relatively-self-contained body of knowledge which usually contains an axiomatic system and all its derived theorems. An axiomatic system that is completely described is a special kind of formal system. A formal theory is an axiomatic system (usually formulated within model theory) that describes a set of sentences that is closed under logical implication.^{[1]} A formal proof is a complete rendition of a mathematical proof within a formal system.

## Properties

An axiomatic system is said to be *consistent* if it lacks contradiction. That is, it is impossible to derive both a statement and its negation from the system's axioms. Consistency is a key requirement for most axiomatic systems, as the presence of contradiction would allow any statement to be proven (principle of explosion).

In an axiomatic system, an axiom is called *independent* if it cannot be proven or disproven from other axioms in the system. A system is called independent if each of its underlying axioms is independent. Unlike consistency, independence is not a necessary requirement for a functioning axiomatic system — though it is usually sought after to minimize the number of axioms in the system.

An axiomatic system is called *complete* if for every statement, either itself or its negation is derivable from the system's axioms (equivalently, every statement is capable of being proven true or false).^{[2]}

## Relative consistency

Beyond consistency, relative consistency is also the mark of a worthwhile axiom system. This describes the scenario where the undefined terms of a first axiom system are provided definitions from a second, such that the axioms of the first are theorems of the second.

A good example is the relative consistency of absolute geometry with respect to the theory of the real number system. Lines and points are undefined terms (also called primitive notions) in absolute geometry, but assigned meanings in the theory of real numbers in a way that is consistent with both axiom systems.^{[citation needed]}

## Models

A

^{[clarification needed}

*abstract model*which is based on other axiomatic systems.

Models can also be used to show the independence of an axiom in the system. By constructing a valid model for a subsystem without a specific axiom, we show that the omitted axiom is independent if its correctness does not necessarily follow from the subsystem.

Two models are said to be isomorphic if a one-to-one correspondence can be found between their elements, in a manner that preserves their relationship.^{[3]} An axiomatic system for which every model is isomorphic to another is called *categorial* (sometimes *categorical*). The property of categoriality (categoricity) ensures the completeness of a system, however the converse is not true: Completeness does not ensure the categoriality (categoricity) of a system, since two models can differ in properties that cannot be expressed by the semantics of the system.

### Example

As an example, observe the following axiomatic system, based on

- (informally, there exist two different items).

- (informally, there exist three different items).

Informally, this infinite set of axioms states that there are infinitely many different items. However, the concept of an infinite set cannot be defined within the system — let alone the cardinality of such as set.

The system has at least two different models – one is the natural numbers (isomorphic to any other countably infinite set), and another is the real numbers (isomorphic to any other set with the cardinality of the continuum). In fact, it has an infinite number of models, one for each cardinality of an infinite set. However, the property distinguishing these models is their cardinality — a property which cannot be defined within the system. Thus the system is not categorial. However it can be shown to be complete.

## Axiomatic method

Stating definitions and propositions in a way such that each new term can be formally eliminated by the priorly introduced terms requires primitive notions (axioms) to avoid infinite regress. This way of doing mathematics is called the **axiomatic method**.^{[4]}

A common attitude towards the axiomatic method is logicism. In their book *Principia Mathematica*, Alfred North Whitehead and Bertrand Russell attempted to show that all mathematical theory could be reduced to some collection of axioms. More generally, the reduction of a body of propositions to a particular collection of axioms underlies the mathematician's research program. This was very prominent in the mathematics of the twentieth century, in particular in subjects based around homological algebra.

The explication of the particular axioms used in a theory can help to clarify a suitable level of abstraction that the mathematician would like to work with. For example, mathematicians opted that rings need not be commutative, which differed from Emmy Noether's original formulation. Mathematicians decided to consider topological spaces more generally without the separation axiom which Felix Hausdorff originally formulated.

The

### History

Mathematical methods developed to some degree of sophistication in ancient Egypt, Babylon, India, and China, apparently without employing the axiomatic method.

^{[6]}

Many axiomatic systems were developed in the nineteenth century, including

### Issues

Not every consistent body of propositions can be captured by a describable collection of axioms. In recursion theory, a collection of axioms is called

In practice, not every proof is traced back to the axioms. At times, it is not even clear which collection of axioms a proof appeals to. For example, a number-theoretic statement might be expressible in the language of arithmetic (i.e. the language of the Peano axioms) and a proof might be given that appeals to topology or complex analysis. It might not be immediately clear whether another proof can be found that derives itself solely from the Peano axioms.

Any more-or-less arbitrarily chosen system of axioms is the basis of some mathematical theory, but such an arbitrary axiomatic system will not necessarily be free of contradictions, and even if it is, it is not likely to shed light on anything. Philosophers of mathematics sometimes assert that mathematicians choose axioms "arbitrarily", but it is possible that although they may appear arbitrary when viewed only from the point of view of the canons of deductive logic, that appearance is due to a limitation on the purposes that deductive logic serves.

### Example: The Peano axiomatization of natural numbers

The mathematical system of natural numbers 0, 1, 2, 3, 4, ... is based on an axiomatic system first devised by the mathematician Giuseppe Peano in 1889. He chose the axioms, in the language of a single unary function symbol *S* (short for "successor"), for the set of natural numbers to be:

- There is a natural number 0.
- Every natural number
*a*has a successor, denoted by*Sa*. - There is no natural number whose successor is 0.
- Distinct natural numbers have distinct successors: if
*a*≠*b*, then*Sa*≠*Sb*. - If a property is possessed by 0 and also by the successor of every natural number it is possessed by, then it is possessed by all natural numbers ("
*Induction axiom*").

### Axiomatization

In

## See also

*.*

**Axiomatic system**- Axiom schema – Short notation for a set of statements that are taken to be true
- Formalism – View that mathematics does not necessarily represent reality, but is more akin to a game
- Gödel's incompleteness theorems – Limitative results in mathematical logic
- Hilbert-style deduction system– System of formal deduction in logic
- History of logic
- List of logic systems
- Logicism – Programme in the philosophy of mathematics
- Zermelo–Fraenkel set theory – Standard system of axiomatic set theory, an axiomatic system for set theory and today's most common foundation for mathematics.

## References

**^**Weisstein, Eric W. "Theory".*mathworld.wolfram.com*. Retrieved 2019-10-31.**^**Weisstein, Eric W. "Complete Axiomatic Theory".*mathworld.wolfram.com*. Retrieved 2019-10-31.**^**Hodges, Wilfrid; Scanlon, Thomas (2018), "First-order Model Theory", in Zalta, Edward N. (ed.),*The Stanford Encyclopedia of Philosophy*(Winter 2018 ed.), Metaphysics Research Lab, Stanford University, retrieved 2019-10-31**^**"*Set Theory and its Philosophy, a Critical Introduction*S.6; Michael Potter, Oxford, 2004**^**Weisstein, Eric W. "Zermelo-Fraenkel Axioms".*mathworld.wolfram.com*. Retrieved 2019-10-31.**^**Lehman, Eric; Meyer, Albert R; Leighton, F Tom.*Mathematics for Computer Science*(PDF). Retrieved 2 May 2023.

## Further reading

- "Axiomatic method",
*Encyclopedia of Mathematics*, EMS Press, 2001 [1994] - Eric W. Weisstein,
*Axiomatic System*, From MathWorld—A Wolfram Web Resource. Mathworld.wolfram.com & Answers.com