Abstract

Ordinal systems are structures for describing ordinal notation systems, which extend the more predicative approaches to ordinal notation systems, like the Cantor normal form, the Veblen function and the Schütte Klammer symbols, up to the Bachmann-Howard ordinal. σ-ordinal systems, which are natural extensions of this approach, reach without the use of cardinals the strength of the transfinitely iterated fixed theories IDσ in an essentially predicative way. We explore the relationship with the traditional approach to ordinal notation systems via cardinals and determine, using “extended Schütte Klammer symbols”, the exact strength of σ-ordinal systems.

Introduction

Motivation

The original problem, which motivated the research in this article, seemed to be a pedagogical one. Several times we have tried to teach ordinal notation systems above the Bachmann-Howard ordinal. The impression we got was that we were able to teach the technical development of these ordinal notation systems, but that some doubts in the audience always persisted. It remained unclear why one could get a well-ordered notation system by denoting small ordinals by big cardinals.