1. The second recursion theorem can be applied to any total recursive function. 2. The second part of the first recursion theorem follows from the first part. 3. To apply the first recursion theorem , the recursion equations must first be recast as a recursive operator. 4. A classic example using the second recursion theorem is the function Q ( x, y ) = x. 5. Quines are possible in any Turing complete programming language, as a direct consequence of Kleene's recursion theorem . 6. Like the second recursion theorem , the first recursion theorem can be used to obtain functions satisfying systems of recursion equations. 7. Like the second recursion theorem, the first recursion theorem can be used to obtain functions satisfying systems of recursion equations. 8. The same definition of recursive function can be given, in computability theory, by applying Kleene's recursion theorem . 9. Compared to the second recursion theorem , the first recursion theorem produces a stronger conclusion but only when narrower hypotheses are satisfied. 10. Compared to the second recursion theorem, the first recursion theorem produces a stronger conclusion but only when narrower hypotheses are satisfied.
