Hjorth's reflection argument

Abstract

Hjorth, assuming $\mathsf{AD}+\mathsf{ZF}+\mathsf{DC}$, showed that there is no sequence of length ${\omega }_2$ consisting of distinct ${\Sigma }_2^1$-sets. We show that the same theory implies that for $n\ge 0$, there is no sequence of length ${\delta }_{2n+2}^1$ consisting of distinct ${\Sigma }_{2n+2}^1$ sets. The theorem settles Question 30.21 of Kanamori, which was also conjectured by Kechris.

arXiv commentSome typos and minor mistakes were corrected