Hjorth's reflection argument

TL;DR

This paper extends Hjorth's reflection argument under AD+ZF+DC to show the non-existence of long sequences of distinct projective sets at higher levels, settling a question posed by Kanamori and Kechris.

Contribution

It generalizes Hjorth's reflection argument to higher projective levels, establishing new non-existence results for sequences of distinct sets.

Findings

No sequence of length ^1_{2n+2} of distinct ^1_{2n+2} sets exists for n  0.

The result confirms a conjecture by Kechris and answers Kanamori's Question 30.21.

The proof relies on the assumption of AD+ZF+DC.

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.