Difference between revisions of "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 [[theorem]]s being provable. | + | 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. | ||
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Another example is the axiom of [[mathematical induction]]: <math>F(0) \wedge \forall n(F(n) \Longrightarrow F(n + 1)) \Longrightarrow \forall k F(k)</math>. This says that if <math>F(0)</math> is true and <math>F(n)</math> implies <math>F(n + 1)</math> for all <math>n</math>, then <math>F(k)</math> is true for every [[nonnegative]] [[integer]] <math>k</math>. 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). | Another example is the axiom of [[mathematical induction]]: <math>F(0) \wedge \forall n(F(n) \Longrightarrow F(n + 1)) \Longrightarrow \forall k F(k)</math>. This says that if <math>F(0)</math> is true and <math>F(n)</math> implies <math>F(n + 1)</math> for all <math>n</math>, then <math>F(k)</math> is true for every [[nonnegative]] [[integer]] <math>k</math>. 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. | + | In [[geometry]], we sometimes use the word ''postulate'' instead of axiom. |
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Latest revision as of 18:11, 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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