Difference between revisions of "Axiom"
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− | An '''axiom''' is a statement that mathematicians assume to be true. Choosing to assume different axioms leads to different systems of [[mathematical logic | + | An '''axiom''' is a statement that mathematicians assume to be true. Choosing to assume different axioms leads to different systems of [[Mathematics|mathematical logic]] and to different [[theorem]]s being provable. |
For example, the statement <math>a \times b = b \times a</math> for [[real number]]s <math>a</math> and <math>b</math> is an axiom (one of the [[field]] axioms of [[real numbers]]). However, this statement does not hold true for any objects; for [[matrix|matrices]], not only is this not an axiom, it is not true. | For example, the statement <math>a \times b = b \times a</math> for [[real number]]s <math>a</math> and <math>b</math> is an axiom (one of the [[field]] axioms of [[real numbers]]). However, this statement does not hold true for any objects; for [[matrix|matrices]], not only is this not an axiom, it is not true. |
Latest revision as of 19:11, 14 July 2021
An axiom is a statement that mathematicians assume to be true. Choosing to assume different axioms leads to different systems of mathematical logic and to different theorems being provable.
For example, the statement for real numbers
and
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: . This says that if
is true and
implies
for all
, then
is true for every nonnegative integer
. 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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