# On the Uniqueness Problem for Notations of Recursive Ordinals

## TL;DR

This paper addresses the problem of uniquely notating recursive ordinals, developing methods to establish such uniqueness and applying these results to classify computable functions hierarchically.

## Contribution

It introduces new methods for proving the uniqueness of recursive ordinal notations and applies these to the hierarchical classification of computable functions.

## Key findings

- Established conditions for unique notations of recursive ordinals
- Applied results to classify computable functions hierarchically
- Provided insights into non-constructive methods in ordinal notation

## Abstract

In the article 'Ordinal Logics and the Characterizations of the Informal Concept of Proof', Georg Kreisel poses the problem of assigning unique notations to recursive ordinals, and additionally suggests that the methods which are developed for its solution will be non-constructive in character. In this paper we develop methods in which various uniqueness results for notations of recursive ordinals can be obtained, and thereafter apply these results to investigate the problems surrounding the hierarchical classification of the computable functions.