Scales and the fine structure of K(R). Part I: Acceptability above the reals

TL;DR

This paper develops a fine structure theory for the inner model $K(R)$, establishing acceptability above the reals for iterable real premice, which is crucial for analyzing the complexity of scales in subsequent parts.

Contribution

It proves that every iterable real premouse is acceptable above the reals, advancing the understanding of the fine structure of $K(R)$ for scale analysis.

Findings

Proves that iterable real premice are acceptable above the reals.

Establishes foundational results for analyzing scales in $K(R)$.

Supports subsequent work on minimal complexity of scales.

Abstract

This article is Part I in a series of three papers devoted to determining the minimal complexity of scales in the inner model $K(\mathbb{R})$. Here, in Part I, we shall complete our development of a fine structure theory for $K(\mathbb{R})$ which is essential for our work in Parts II and III. In particular, we prove the following fundamental theorem which supports our analysis of scales in $K(\mathbb{R})$: If $\mathcal{M}$ is an iterable real premouse, then $\mathcal{M}$ is acceptable above the reals. This theorem will be used in Parts II and III to solve the problem of finding scales of minimal complexity in $K(\mathbb{R})$.