# Axiom

## Statement that is taken to be true / From Wikipedia, the free encyclopedia

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An **axiom**, **postulate**, or **assumption** is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (*axíōma*), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.[1][2]

The precise definition varies across fields of study. In classic philosophy, an axiom is a statement that is so evident or well-established, that it is accepted without controversy or question.[3] In modern logic, an axiom is a premise or starting point for reasoning.[4]

In mathematics, an *axiom* may be a "logical axiom" or a "non-logical axiom". Logical axioms are taken to be true within the system of logic they define and are often shown in symbolic form (e.g., (*A* and *B*) implies *A*), while non-logical axioms (e.g., *a* + *b* = *b* + *a*) are substantive assertions about the elements of the domain of a specific mathematical theory, such as arithmetic.

Non-logical axioms may also be called "postulates" or "assumptions". In most cases, a non-logical axiom is simply a formal logical expression used in deduction to build a mathematical theory, and might or might not be self-evident in nature (e.g., the parallel postulate in Euclidean geometry). To axiomatize a system of knowledge is to show that its claims can be derived from a small, well-understood set of sentences (the axioms), and there are typically many ways to axiomatize a given mathematical domain.

Any axiom is a statement that serves as a starting point from which other statements are logically derived. Whether it is meaningful (and, if so, what it means) for an axiom to be "true" is a subject of debate in the philosophy of mathematics.[5]