proof of principle of transfinite induction
To prove the transfinite induction theorem, we note that the class of ordinals is well-ordered by . So suppose for some , there are ordinals such that is not true. Suppose further that satisfies the hypothesis, i.e. . We will reach a contradiction.
The class is not empty. Note that it may be a proper class, but this is not important. Let be the -minimal element of . Then by assumption, for every , is true. Thus, by hypothesis, is true, contradiction.
|Title||proof of principle of transfinite induction|
|Date of creation||2013-03-22 12:29:06|
|Last modified on||2013-03-22 12:29:06|
|Last modified by||jihemme (316)|