Algebraic Ordinals

TL;DR

This paper characterizes algebraic ordinals as those less than ^{^{}}, showing a precise boundary for the order types of algebraic trees' frontiers.

Contribution

It establishes that algebraic ordinals are exactly those below ^{^{}}, providing a clear classification of these ordinals.

Findings

Algebraic ordinals are exactly those less than ^{^{}}.

The frontier of algebraic trees forms well-ordered sets with these ordinals.

The order type of the frontier of an algebraic tree is an algebraic ordinal.

Abstract

An algebraic tree T is one determined by a finite system of fixed point equations. The frontier \Fr(T) of an algebraic tree t is linearly ordered by the lexicographic order \lex. When (\Fr(T),\lex) is well-ordered, its order type is an \textbf{algebraic ordinal}. We prove that the algebraic ordinals are exactly the ordinals less than $\omega^{\omega^\omega}$.