Remark. Since every well-ordered set is well-founded, the well-founded recursion theorem is a generalization of the transfinite recursion theorem. Notice that the here is a function in two arguments, and that it is necessary to specify the element in the first argument (in contrast with the in the transfinite recursion theorem), since it is possible that for in a well-founded set.
By converting into a formula ( such that for all , there is a unique such that ), then the above theorem can be proved in ZF (with the aid of the well-founded induction). The proof is similar to the proof of the transfinite recursion theorem.
|Date of creation||2013-03-22 17:54:52|
|Last modified on||2013-03-22 17:54:52|
|Last modified by||CWoo (3771)|