Universal sets for pointsets properly on the nth level of the projective hierarchy

Abstract

The Axiom of Projective Determinacy implies the existence of a universal $\underset{\tilde{}}{{\Pi}^1_n}\setminus\underset{\tilde{}}{{\Delta}^1_n}$ set for every $n \geq 1$. Assuming $\rm{MA}(\aleph_{1})+\aleph_{1}=\aleph_{1}^{\mathbb{L}}$ there exists a universal $\underset{\tilde{}}{{\Pi}^1_1}\setminus\underset{\tilde{}}{{\Delta}^1_1}$ set. In ZFC there is a universal $\underset{\tilde{}}{{\Pi}^0_\alpha}\setminus\underset{\tilde{}}{{\Delta}^0_\alpha}$ set for every $\alpha$.