proof of Zermelo’s well-ordering theorem
(the function is undefined if either of the unless clauses holds).
Thus is just (the least element of ), and (the least element of other than ).
Since the ordinals are well ordered, there is a least ordinal not in , and therefore is undefined. It cannot be that the second unless clause holds (since is the least such ordinal) so it must be that , and therefore for every there is some such that . Since we already know that is injective, it is a bijection between and , and therefore establishes a well-ordering of by .
|Title||proof of Zermelo’s well-ordering theorem|
|Date of creation||2013-03-22 12:59:07|
|Last modified on||2013-03-22 12:59:07|
|Last modified by||Henry (455)|