Covering of ordinals

TL;DR

This paper explores the structure of fundamental sequences of ordinals below , providing a logical characterization, classifying these structures within the pushdown hierarchy, and locating ordinals within this framework.

Contribution

It constructs a monadic second-order formula for structures of ordinals below  and classifies these structures in the pushdown hierarchy, offering new insights into ordinal representations.

Findings

A monadic second-order formula identifies structures of ordinals below .

Structures are classified within the pushdown hierarchy.

Ordinals are explicitly located in the hierarchy with a direct presentation.

Abstract

The paper focuses on the structure of fundamental sequences of ordinals smaller than $\epsilon_0$. A first result is the construction of a monadic second-order formula identifying a given structure, whereas such a formula cannot exist for ordinals themselves. The structures are precisely classified in the pushdown hierarchy. Ordinals are also located in the hierarchy, and a direct presentation is given.