Axioms

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Axioms are statements accepted as true without proofs. Adding a proof to an axiom turns it into a theorem.

Examples of axioms include:

• Any two points on a plane can be connected by a line.
• There are infinitely many positive integers.
• If $a=b$ and $b=c$, then $a=c$ also.

Not all axioms can be turned into theorems. For example, many people have tried to prove Euclid's fifth postulate (the "parallel postulate").

Axioms should not be confused with statements that are a matter of definition. For example, the statement "no positive prime number is divisible by a different positive prime number," although true, is not an axiom, as it is a consequence of the definition of prime number: if $p\neq q$ and $q|p$, then $p$ is not a prime number.