Determinacy in L(R,\mu)

TL;DR

This paper characterizes the conditions under which the model L(ℝ,μ) satisfies certain determinacy axioms, linking large cardinal assumptions with descriptive set theory in a fine-structural setting.

Contribution

It provides a characterization theorem for L(ℝ,μ), establishing equivalence between Θ>ω₂ and AD⁺, and shows equiconsistency between large cardinals and determinacy assumptions.

Findings

L(ℝ,μ) satisfies Θ>ω₂ iff it satisfies AD⁺.

Equiconsistency between ZFC with ω² Woodin cardinals and ZF+DC with a normal fine measure and Θ>ω₂.

Theorem linking large cardinal hypotheses with determinacy in models of set theory.

Abstract

Assume L(\mathbb{R},\mu) satisfies ZF+DC+\Theta>\omega_2 + \mu is a normal fine measure on \powerset_{\omega_1}(\mathbb{R}). The main result of this paper is the characterization theorem of L(\mathbb{R},\mu) which states that L(\mathbb{R},\mu) satisfies \Theta>\omega_2 if and only if L(\mathbb{R},\mu) satisfies AD^+. As a result, we obtain the equiconsistency between the two theories: "ZFC + there are \omega^2 Woodin cardinals" and "ZF+DC+\mu is a normal fine measure on \powerset_{\omega_1}(\mathbb{R}) + \Theta>\omega_2".