FOM: aksiom of constructibility

[Kanovei]

A typical objection against V=L is that it restricts 
the multitude of sets. 
If one draws a parallel with standard/nonstandard 
mathematics, then the same argument will work against 
the "standard" model of PA and in favour of a suitable 
choice of a nonstandard model 
(say, recursively saturated) as the base for arithmetic, 
which indeed does not happen. 

Still there is an interesting fom-like question related 
to V=L: 

QUESTION: is there any mathematically meaningful statement 
about projective sets 
(in particular different from consistency of something) 
which V=L does not decide ? 

Vladimir Kanovei