## Axiom |

An **axiom** or **postulate** is a statement that is taken to be *axíōma* (ἀξίωμα) 'that which is thought worthy or fit' or 'that which commends itself as evident.'^{[1]}^{[2]}

The term has subtle differences in definition when used in the context of different fields of study. As defined in ^{[3]} As used in modern ^{[4]}

As used in *axiom* is used in two related but distinguishable senses: "logical axioms" and "non-logical axioms". Logical axioms are usually statements that 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 actually substantive assertions about the elements of the domain of a specific mathematical theory (such as

When used in the latter sense, "axiom", "postulate", and "assumption" may be used interchangeably. 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., ^{[5]} 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 may be multiple 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 ^{[6]}

The word *axiom* comes from the *axíōma*), a *axioein*), meaning "to deem worthy", but also "to require", which in turn comes from ἄξιος (*áxios*), meaning "being in balance", and hence "having (the same) value (as)", "worthy", "proper". Among the ^{[7]}

The root meaning of the word *postulate* is to "demand"; for instance, ^{[8]}

Ancient geometers maintained some distinction between axioms and postulates. While commenting on Euclid's books, ^{[9]} *petitio* and called the axioms *notiones communes* but in later manuscripts this usage was not always strictly kept.