What is the simplest non-constructible set of integers (say, in the analytical hierarchy) that is compatible with the nonexistence of $0^\sharp$? In particular, can there still be a non-constructible $\Delta^1\_3$ set of integers if one assumes the nonexistence of $0^\sharp$?



Simplest non-constructible set of integers compatible with the nonexistence of $0^\sharp$?
It is consistent, relative to just $ZFC$, that there exists a non-constructible $\Delta\_3^1$ set of integers.
Such a result was first produced in the paper ``Some allpications of almost disjoint sets'' by Jensen-Solovay, using almost disjoint forcing.


In fact we can prove the following stronger result which is due to Jensen ``definable sets of minimal degree'':


It is consistent that $V=L[R],$ where $R$ is minimal (in the degree of constructibility) and it is the unique solution of a $\Pi\_2^1$ predicate (hence it is $\Delta\_3^1$). 


