An axiomatic approach to the ordinal number system

Abstract

ENGLISH ABSTRACT: Ordinal numbers are transfinite generalisations of natural numbers, and are usually defined and studied concretely as special types of sets. In this thesis we explore an abstract approach to developing the theory of ordinal numbers, where we present various axiomatisations of an ordinal number system and prove their equivalence. Since ordinal numbers do not form a set, in order to develop such a theory one needs to extend the usual framework of Zermelo-Fraenkel set theory. Among several such possible extensions, we pick the one that is based on the notion of a Grothendieck universe. While some of the results obtained in this thesis are merely adaptations of known results to this context, some others are new even to classical set theory. Among these is a definition and a universal property of the ordinal number system that mimics the classical Dedekind-Peano approach to the natural number system.

AFRIKAANSE OPSOMMING: Ordinaalgetalle is transfiniete veralgemenings van die telgetalle, en word gewoonlik konkreet gedefinieer en bestudeer as spesiale soorte stelle. In hierdie tesis ondersoek ons ’n abstrakte benadering tot die ontwikkeling van ordinaalgetalteorie, waarin ons verskeie aksiomatiserings van ordinaalgetalstelsels gee, en hul ekwivalensie bewys. Aangesien ordinaalgetalle nie ’n stel vorm nie, is dit nodig om die standaard raamwerk van Zermelo-Fraenkel stelteorie uit te brei om so ’n teorie te kan ontwikkel. Vanuit verskeie moontlike raamwerke kies ons een wat op die idee van ’n Grothendieck universum gebaseer is. Alhoewel sommige van die bevindings in hierdie tesis slegs aanpassings van bekende bevindings na hierdie konteks is, is ander nuut selfs in klassieke stelteorie. Die nuwe bevindings sluit ’n definisie en universele eienskap van die ordinaalgetalstelsel in, wat die klassieke Dedekind-Peano benadering tot die telgetalstelsel naboots.