Note on $\mathsf{TD} + \mathsf{DC}_\mathbb{R}$ implying $\mathsf{AD}^{L(\mathbb{R})}$

TL;DR

This paper provides a concise core model induction proof demonstrating that the Axiom of Determinacy for L(R) follows from the assumptions of the Tree Property plus the Dependent Choice over the reals.

Contribution

It introduces a streamlined core model induction argument establishing $ ext{AD}^{L( ext{R})}$ from $ ext{TD} + ext{DC}_ ext{R}$ assumptions.

Findings

Proves $ ext{AD}^{L( ext{R})}$ from $ ext{TD} + ext{DC}_ ext{R}$

Simplifies the core model induction proof process

Strengthens the connection between tree properties and determinacy in inner models.

Abstract

A short core model induction proof of $\mathsf{AD}^{L(\mathbb{R})}$ from $\mathsf{TD} + \mathsf{DC}_\mathbb{R}$.