# Definable surjections in $J$-structures

Recall that a $J$-structure is an amenable structure of the form ($J_{\alpha}^A,B$) where $A$ and $B$ are predicates and $\alpha$ is a limit ordinal. Then if you let $M=J_{\alpha}^A$, there is a surjective function $f:[\alpha]^{\lneq \omega} \rightarrow M$ which is $\Sigma_1^M$. Can we prove that there exists surjective functions $f:[\alpha]^{\lneq \omega} \rightarrow M$ which are $\Sigma_n^M$ for all $n$<$\omega$?

The proof that there is a surjective function $f:[\alpha]^{\lneq \omega} \rightarrow M$ which is $\Sigma_1^M$ uses that $h_M(\alpha)$ is a $\Sigma_1$ elementary substructure of $M$ ($h_M(\alpha)$ the Skolem hull of $\alpha$)

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I don't understand the question. A $\Sigma^M_1$ function is also (trivially) $\Sigma^M_n$ for all larger $n$. So you seem to already have exactly what you asked for --- unless you really want the case $n=0$. – Andreas Blass Aug 21 '10 at 8:58
That is true, I think I got confused by this small thing. Thanks – Carlo Von Schnitzel Aug 21 '10 at 9:01

A $\Sigma^M_1$ function is also (trivially) $\Sigma^M_n$ for all larger $n$.