Axiom

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An axiom is a statement that mathematicians assume true. Choosing different axioms leads to different systems of mathematical logic and to different theorems being provable.

For example, the statement $a \times b = b \times a$ for real numbers $a$ and $b$ is an axiom (one of the field axioms of real numbers). However, this statement does not hold true for any objects; for matrices, not only is this not an axiom, it is not true.

Another example is the axiom of mathematical induction: $F(0) \wedge \forall n(F(n) \Longrightarrow F(n + 1)) \Longrightarrow \forall k F(k)$ . This says that if $F(0)$ is true and $F(n)$ implies $F(n + 1)$ for all $n$ , then $F(k)$ is true for every nonnegative integer $k$ . This statement is not something we prove. Rather, it is something that we assume to be true about the integers in order to prove other statements. It is possible to develop different axiomatizations of arithmetic which lack the axiom of mathematical induction, but these are generally much weaker systems (that is, fewer statements are provable).

In geometry, we sometimes use the word postulate instead of axiom.

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Axiom

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