Definition:Axiom
Contents |
General Definition
In all contexts, the definition of the term axiom is by and large the same.
That is, an axiom is a statement which is accepted as being true.
An alternative term for it is postulate.
A statement that is considered an axiom can be described as being axiomatic.
A logical or mathematical system which possesses an explicitly stated set of axioms from which theorems can be derived is called an axiomatic system.
Logic
An axiom in logic is a statement which is taken as self-evident.
Note, however, that there has been disagreement for as long as there have been logicians and philosophers as to whether particular statements are true or not.
For example, the Law of Excluded Middle is accepted as axiomatic by philosophers and logicians of the Aristotelian school but is denied by the intuitionist school.
Formal Systems
Let $\mathcal F$ be a formal system consisting of a formal language with deductive apparatus $\mathcal D$.
An axiom of $\mathcal F$ is a well-formed word in $\mathcal F$ belongs to $\mathcal D$ by definition.
Mathematics
The term axiom is used throughout the whole of mathematics to mean a statement which is accepted as true for that particular branch.
Different fields of mathematics usually have different sets of statements which are considered as being axiomatic.
So statements which are taken as axioms in one branch of mathematics may be theorems, or irrelevant, in others.
It is possible (and this is the ultimate aim of ProofWiki) to justify basing the whole of mathematics on a handful of axioms.
Also see
