Carlson's $<_1$-relation on the class of additive principal ordinals

TL;DR

This paper introduces the study of Carlson's <_1-relation on all ordinals, focusing on isomorphisms and the <^0-relation, to understand the structure of thin k-club classes of ordinals.

Contribution

It provides the first detailed analysis of Carlson's <_1-relation on additive principal ordinals and explores its connection with isomorphisms and club classes.

Findings

Analysis of isomorphisms g(0,a,b) among additive principal ordinals

Relationship between <_1 and the <^0-relation

Indication of how <_1 induces thinner k-club classes

Abstract

This is the first in a series of several articles. Our general purpose is to investigate Carlson's <_1-relation in the whole class of ordinals and later link it with ordinals of proof-theoretic interests. In this introductory article, after giving the basic definitions, we study the (canonical) isomorphisms {g(0,a,b)|a,b are additive principal numbers} and the <^0-relation and see how it is that <_1 induces, through all of these notions, thinner k-club classes of ordinals. In coming articles it will be shown the complete generalization of these ideas to the thinnest k-club classes induced by <_1.