My adviser is out of town and there is a comment in the Van Wesep paper "wadge degrees and descriptive set theory" that I can't figure out.

Work in ZF+AD throughout.

As stated in the title, the comment says that for n odd, the (order type of the) $\bf\Delta^1_n$ wadge degrees are less that $\bf\delta^1_{n+1}$ and Van Wesep includes the comment "(Prewellorder the codes of of $\bf\Delta^1_n$ sets as preimages of initial segments of a $\bf\Pi^1_n$-prewellordered complete $\bf\Pi^1_n$ set)".

This is easy enough, let U be some $\bf\Pi^1_n$ complete set with corresponding well-order $\phi$. For x and y codes of $\bf\Delta^1_n$ sets, let $x\leq y$ iff $\exists w \in U_x$ such that (x,w) is $\leq_\phi$ greater than all (y,z) such that $z\in U_y$. This will be defined for all such x,y because of the boundedness principle.

Now, in order to finish the proof, one must show that for $\bf\Delta^1_n$ codes x,y, $U_y <_w U_x \rightarrow y < x$. While this seems intuitively true (if $y<x$ we can compute $U_y$ from $U_x$ using $<_\phi$), I cannot prove this fact.

I believe this is the correct approach. One alternative way of defining $\leq$ would be $y\leq x$ iff $(<_\phi |U_y) \leq$ (as an order type) $ (<_\phi |U_x)$ or something along those lines. But, this seems less hopeful.

Any help would be greatly appreciated. I apologize if this question is too basic for this website.

,

Cody