This is an example of a construction by transfinite recursion.
2.
The constructible hierarchy, L is defined by transfinite recursion.
3.
This is again a definition by transfinite recursion.
4.
? 1 1-CA 0 is stronger than arithmetical transfinite recursion and is fully impredicative.
5.
Such a definition is normally said to be by transfinite recursion the proof that the result is well-defined uses transfinite induction.
6.
More generally, recursive definitions of functions can be made whenever the domain is a well-ordered set, using the principle of transfinite recursion.
7.
Each can be defined in essentially two different ways : either by constructing an explicit well-ordered set that represents the operation or by using transfinite recursion.
8.
It being understood that each parameter's possible values are ordered according to the restriction of the ordering of L to L ?, so this definition involves transfinite recursion on ?.
9.
This definition assumes the " F " ( ? ) known in the very process of defining " F "; this apparent vicious circle is exactly what definition by transfinite recursion permits.
10.
In a sense, ? 1 1-CA 0 comprehension is to arithmetical transfinite recursion ( ? 1 1 separation ) as ACA 0 is to weak K�nig's lemma ( ? 0 1 separation ).
