Universal sets for pointsets properly on the n th level of the projective hierarchy

Journal of Symbolic Logic 78 (1):237-244 (2013)
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Abstract

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

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Set Theory.Keith J. Devlin - 1981 - Journal of Symbolic Logic 46 (4):876-877.
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Descriptive Set Theory.Yiannis Nicholas Moschovakis - 1982 - Studia Logica 41 (4):429-430.
Long Projective Wellorderings.Leo Harrington - 1977 - Annals of Mathematical Logic 12 (1):1.

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