The real numbers in inner models of set theory

TL;DR

This paper explores the structure and properties of real numbers within various inner models of set theory, extending classical results and introducing new concepts like infinite order gaps to deepen understanding of their generation and irregularities.

Contribution

It generalizes existing results on reals in the constructible universe, introduces the concept of infinite order gaps, and extends analysis to models like L[#_1] and L[#], providing new insights.

Findings

Detailed proofs and generalizations of gaps in L

Introduction of infinite order gaps concept

Analysis of reals in models L[#_1] and L[#]

Abstract

We study the structural regularities and irregularities of the reals in inner models of set theory. Starting with $L$, G\"{o}del's constructible universe, our study of the reals is thus two-fold. On the one hand, we study how their generation process is linked to the properties of $L$ and its levels, mainly referring to "Gaps in the constructible universe" (Marek and Srebrny, 1974). We provide detailed proofs for the results of that paper, generalize them in some directions hinted at by the authors, and present a generalization of our own by introducing the concept of an infinite order gap, which is natural and yields some new insights. On the other hand, we present and prove some well-known results that build pathological sets of reals. We generalize this study to $L[\#_1]$ (the smallest inner model closed under the sharp operation for reals) and $L[\#]$ (the smallest inner model closed under all sharps), for which we provide some introduction and basic facts which are not easily available in the literature. We also discuss some relevant modern results for bigger inner models.