well-ordering principle for natural numbers proven from the principle of finite induction
We will use the principle of finite induction (the strong form) to show that is empty, a contradition.
Fix any natural number , and suppose that for all natural numbers , . If , then (*) implies that there is an element such that . This would be incompatible with the assumption that for all natural numbers , . Hence, we conclude that is not in .
Therefore, by induction, no natural number is a member of . The set is empty.
|Title||well-ordering principle for natural numbers proven from the principle of finite induction|
|Date of creation||2013-03-22 16:38:02|
|Last modified on||2013-03-22 16:38:02|
|Last modified by||CWoo (3771)|