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

Abstract

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

Theorems & Definitions (170)

• Definition 1.1
• Theorem 1.2: Harrington
• Theorem 1.3: Kechris
• Theorem 1.4: Jackson
• Theorem 1.5: Jackson
• Theorem 1.6: Hjorth
• Corollary 1.7
• Theorem 1.8: Sargsyan
• Theorem 1.9: Kechris
• Conjecture 1.10: Sargsyan