wrong theorem
Kleene’s amazing second recursion theorem by Moschovakis contains:
A list of some (18) of the most significant applications of the Kleene's Second Recursion Theorem, in a kind of “retrospective exhibition” of the work that it has done since 1938. [...] These were selected because of their foundational significance and the variety of ways in which they witness how the recursion theorem is used.
A talk by Moschovakis gives an overview.