Q:

Ordinal definable sets of reals in the Solovay

To be precise, let $\Omega$ be an inaccessible cardinal in $L$ and let N be the Solovay model defined by the Levy-collapse in this case.  Then $\Omega$ is $\aleph_1$ in $N$. 
How many different OD (=ordinal-definable) sets of reals are there in $N$?  
One answers that there is exactly $\aleph_2$ of them. 
Now, is there a concrete, meaningful OD sequence of $\aleph_2$ different OD sets of reals in $N$? 

A:

We can start with a canonical sequence of $\Omega^+$ many distinct subsets of $\Omega$ in $L$, which is a sequence of $\aleph_2$ many distinct subsets of $\aleph_1$ in $N$.
Then we can use the canonical partial surjection $\mathbb{R} \to \aleph_1$, which takes a real coding a well-ordering of $\omega$ to the order type of this well-ordering, to get a sequence of $\aleph_2$ many distinct subsets of $\mathbb{R}$ in $N$.
I think this example is about as concrete and meaningful as we can hope for, because we will need to use somehow the fact that the ground model satisfies $2^\Omega = \Omega^+$.

