Unreachability of Inductive-Like Pointclasses in $L(\mathbb{R})$

Abstract

Hjorth proved from $ZF+AD+DC$ that there is no sequence of distinct $\hat{I}{\pounds }_2^1$ sets of length $\hat{I}{´}_2^1$. Sargsyan extended Hjorth's technique to show there is no sequence of distinct $\hat{I}{\pounds }_{2n}^1$ sets of length $\hat{I}{´}_{2n}^1$. Sargsyan conjectured an analogous property is true for any regular Suslin pointclass in $L(R)$ -- i.e. if $\hat{I}º$ is a regular Suslin cardinal in $L(R)$, then there is no sequence of distinct $\hat{I}º$-Suslin sets of length $\hat{I}{º}^{+}$ in $L(R)$. We prove this in the case that the pointclass $S(\hat{I}º)$ is inductive-like.