Generic Generators

TL;DR

This paper introduces two principles, Direct Limit Independence and Bounded Direct Limits, that together imply Hod Pair Capturing, addressing a major open problem in descriptive inner model theory.

Contribution

The paper proposes a novel approach by formulating two principles that collectively imply Hod Pair Capturing, advancing understanding in inner model theory.

Findings

The principles imply HPC under certain conditions.

The approach offers a new framework for tackling open problems in descriptive inner model theory.

Potential implications for the development of inner model theory.

Abstract

The goal of this paper is to present an approach to Hod Pair Capturing (HPC). $HPC$ is the most outstanding open problem of descriptive inner model theory. More specifically, we introduce two principles, the Direct Limit Independence and the Bounded Direct Limits, and show that they together imply HPC.