Hjorth's reflection argument

Abstract

Hjorth, assuming ${\sf{AD+ZF+DC}}$, showed that there is no sequence of length $\omega_2$ consisting of distinct $\Sigma^1_2$-sets. We show that the same theory implies that for $n\geq 0$, there is no sequence of length $\delta^1_{2n+2}$ consisting of distinct $\Sigma^1_{2n+2}$ sets. The theorem settles Question 30.21 of Kanamori, which was also conjectured by Kechris.