Ordinal notation systems corresponding to Friedman's linearized well-partial-orders with gap-condition

TL;DR

This paper explores whether addition-free theta functions serve as a canonical notation system for Friedman's well-partial-orders with gap-condition, revealing they work for two labels but not more, and analyzing their order types.

Contribution

It demonstrates the limitations of addition-free theta functions as a canonical notation system for certain well-partial-orders and characterizes their order types in relation to ordinals below .

Findings

Addition-free theta functions form a canonical notation system for two labels.

They do not serve as such for more than two labels.

The order type of these systems is characterized in terms of ordinals less than .

Abstract

In this article we investigate whether the addition-free theta functions form a canonical notation system for the linear versions of Friedman's well-partial-orders with the so-called gap-condition over a finite set of labels. Rather surprisingly, we can show this is the case for two labels, but not for more than two labels. To this end, we determine the order type of the notation systems for addition-free theta functions in terms of ordinals less than $\varepsilon_0$. We further show that the maximal order type of the Friedman ordering can be obtained by a certain ordinal notation system which is based on specific binary theta functions.