Hjorth's reflection argument
Grigor Sargsyan
TL;DR
Under $ZF+AD+DC$, the paper extends Hjorth's reflection-based obstruction from $n=0$ to all $n\, ext{with}\, oldsymbol{oldsymbol{ abla}}^1_{2n+2}$-levels, showing there is no sequence of length $oldsymbol{oldsymbol{ heta}}^1_{2n+2}$ consisting of pairwise distinct $oldsymbol{ extSigma}^1_{2n+2}$-sets. The proof harnesses inner-model theory, employing directed systems of mice and extender algebras, along with Hjorth's reflection technique, to derive a contradiction to any hypothetical long sequence. It also strengthens the understanding of Harrington-type phenomena in the projective hierarchy, yielding corollaries about thin $oldsymbol{oldsymbol{ extPi}}^1_{2n+2}$-equivalence relations and class bounds such as $oldsymbol{oldsymbol{ extdelta}}^1_{2n+1}$ for certain cases under $AD^{L({ eals})}$. The methods avoid prior dependence on Kechris-Martin, illustrating a robust inner-model framework for addressing global questions in the projective hierarchy.
Abstract
Hjorth, assuming ${\sf{AD+ZF+DC}}$, showed that there is no sequence of length $ω_2$ consisting of distinct $Σ^1_2$-sets. We show that the same theory implies that for $n\geq 0$, there is no sequence of length $δ^1_{2n+2}$ consisting of distinct $Σ^1_{2n+2}$ sets. The theorem settles Question 30.21 of Kanamori, which was also conjectured by Kechris.