Pure $\Sigma_2$-Elementarity beyond the Core

TL;DR

This paper presents a detailed coding of the structure ${ m extbf{R}}_2$ that captures $oldsymbol{ m extbf{ extSigma}}_1$- and $oldsymbol{ m extSigma}_2$-elementarity on ordinals, facilitating the study of higher $oldsymbol{ m extSigma}_3$-elementary substructures.

Contribution

It introduces a comprehensive coding of ${ m extbf{R}}_2$ to analyze $oldsymbol{ m extSigma}_2$-elementarity beyond the core, enabling new insights into $oldsymbol{ m extSigma}_3$-elementary structures.

Findings

Complete coding of ${ m extbf{R}}_2$ structure for $oldsymbol{ m extSigma}_1$ and $oldsymbol{ m extSigma}_2$-elementarity.

Framework for analyzing pure $oldsymbol{ m extSigma}_3$-elementary substructures.

Enhanced understanding of the hierarchy of elementary substructures.

Abstract

We display the entire structure ${\cal R}_2$ coding $\Sigma_1$- and $\Sigma_2$-elementarity on the ordinals. This will enable the analysis of pure $\Sigma_3$-elementary substructures.