Notice that the theorem above is not provable in ZF set theory, as and are both classes, not sets. In order to prove this statement, one way of getting around this difficulty is to convert both and into formulas, and modify the statement, as follows:
Let be a formula such that
where means “ is an ordinal”,
is the formula “ is a function”,
is the formula “”,
is the formula , and
is the formula .
A stronger form of the transfinite recursion theorem says:
Let be any formula (in the language of set theory). Then the following is a theorem: assume that satisfies property that, for every , there is a unique such that . If be a well-ordered set (well-ordered by ), then there is a unique function defined on such that
for every . Here, , the initial segment of in .
The above theorem is actually a collection of theorems, or known as a theorem schema, where each theorem corresponds to a formula. The other difference between this and the previous theorem is that this theorem is provable in ZF, because the domain of the function is now a set.
|Date of creation||2013-03-22 17:53:51|
|Last modified on||2013-03-22 17:53:51|
|Last modified by||CWoo (3771)|