Basis theorem (due to Solovay?)

Question

I'm finishing up my bibliography and I'm looking for a reference for the statement that, working in $L(\mathbb{R})$, the $\Delta^2_1$ sets form a basis for the $\Sigma^2_1$ predicates. I believe that it is due to Solovay.

Answer 1

The result is indeed Solovay's Basis Theorem.

It is a consequence of Moschovakis's Coding Lemma, and sometimes it is referred to as (a version of) the reflection theorem (for example, in section 8 of Steel's Handbook article). Perhaps the optimal reference (it is self-contained, and easily accessible) is Section 2.4 of

Peter Koellner, and W. Hugh Woodin. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010.

I do not think it appears explicitly in a paper by Solovay. It should date back to 1976 at the latest, which is when Kechris and Solovay observed that not all $\Pi^2_1$ sets can be uniformized if $V=L(\mathbb R)$ and choice fails (see section 30 in Kanamori's book).