[FOM] A basis for the projective collections of sets

[Fredrik Engström]

Hi!

I am wondering if the set of projective sets is a basis for the 
projective collections of sets, i.e., if P = the union of lightface 
\Sigma^1_k is a basis for itself.

I understand that under the assumption of V=L or PD, it is, since then
\Delta^1_{2k} is a basis for \Sigma^1_{2k} (for k>0). What can ZFC prove
along these lines?

I would very much appreciate references for discussions along those
lines.

Best regards,
Fredrik Engström