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

TL;DR

This paper proves a conjecture about the unreachability of certain sequences of sets in the constructible universe relative to the reals, specifically for inductive-like pointclasses, extending previous results on projective sets.

Contribution

It establishes the unreachability of sequences of distinct sets of a regular Suslin pointclass in $L(R)$ when the pointclass is inductive-like, confirming a conjecture for this class.

Findings

No sequence of distinct $ heta$-Suslin sets of length $ heta^+$ exists in $L(R)$ for inductive-like pointclasses.

Extends known results from projective sets to a broader class of pointclasses.

Supports the conjecture for regular Suslin pointclasses in $L(R)$.

Abstract

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