Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Theorem 1.7.5 on p.35 of Gao's Invariant Descriptive Set Theory reads

Theorem 1.7.5 (Kleene)
If $A\subseteq X \times \omega^{\omega}$ is $\Pi^{1}_{1}$ and $$x \in B \Longleftrightarrow \exists y \in \Delta^{1}_{1}(x)\; (x,y) \in A,$$ then $B$ is also $\Pi^{1}_{1}$.

Here $X = \omega^{m} \times (\omega^\omega)^n$ for some $m$ and $n$ (or equivalently, $X=\omega^\omega$, I suppose).

Where can I find a proof of this result? Feel free to just prove it here.

I checked the bibliography, and, though it seems impossible, there are no references to Kleene; it goes straight from "Kelly" to "Louveau".

share|improve this question
add comment

1 Answer 1

up vote 2 down vote accepted

This is the Socalled Spector-Gandy's theorem. A proof can be found in higher recursion theory by Sacks or descriptive set theory by Moschovakis.

share|improve this answer
1  
Moschovakis's book is available on his homepage –  t.b. May 5 '12 at 16:25
    
Thanks a lot, Liang. –  Quinn Culver May 6 '12 at 13:13
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.