So there is this exercise in Ralf Schindler's book, which I have problems with and I would appreciate any hints or solutions for it. Also here, $\tilde{\Sigma}_1^{J_\alpha[E]}$ denotes the set of $\Sigma_1$-definable relations with parameters from $J_\alpha[E]$. Also the indices of the $J$-hierarchy here are limit ordinals.
Exercise. Let $M = J_\alpha[E]$ be a $J$-structure. Show that there is a (partial) surjection $h:\alpha\rightarrow [\alpha]^{\lt\omega}$ such that $h \in \tilde{\Sigma}_1^{M}$.
My thoughts:
I tried using different methods to construct this. First because of the $\Sigma_1$-definability clause I tried building an order on $[\alpha]^{\lt\omega}$, and letting $h$ compute the order-type. But the order I used was the usual ordering on finite subsets of ordinals(i.e. $u \lt v \leftrightarrow \max(u \Delta v) \in v$) and may have large order-types below $[\alpha]^{\lt\omega}$.
Then my next idea was to prove this inductively. First we observe that for $\alpha = \omega$, we can use the order-type idea above. For $\alpha = \beta + \omega$, we have a function $h_\beta:\beta\rightarrow [\beta]^{\lt\omega}$ such that $h_\beta \in \tilde{\Sigma}_1^{J_\beta[E]}$ and so $h_\beta \in J_\alpha[E]$ and so we can use it as a parameter in our formula. Now it suffices to extend $h_\beta$ in a $\Sigma_1$ way over $\alpha$ while assuring surjectivity on $[\alpha]^{\lt\omega}$. I did this by doing some codings and I have no problem here. My main issue is where $\alpha$ is a limit of limit ordinals. In that case we can't choose functions as parameters to extend, so I guess it must be done directly.
So my question is: Can my above argument be completed? Or is there some other direct way to prove this?
You might consider finding a surjection onto all of $M$ itself, then just restricting to the preimage of $[\alpha]^{<\omega}$ (this is why Schindler notes that the function is partial). The result is Lemma 2.10 of Jensen's original paper, which can found in full here, where the result is proven.
I unfortunately don't have time to summarize the proof right now, but that should help point you in the right direction.