# An axiomatic approach to the ordinal number system

 dc.contributor.advisor Janelidze, Zurab en_ZA dc.contributor.author Van der Berg, Ineke en_ZA dc.contributor.other Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Mathematics. en_ZA dc.date.accessioned 2021-03-06T12:36:30Z dc.date.accessioned 2021-04-21T14:31:07Z dc.date.available 2021-03-06T12:36:30Z dc.date.available 2021-04-21T14:31:07Z dc.date.issued 2021-03 dc.identifier.uri http://hdl.handle.net/10019.1/109901 dc.description Thesis (MSc)--Stellenbosch University, 2021. en_ZA dc.description.abstract ENGLISH ABSTRACT: Ordinal numbers are transfinite generalisations of natural numbers, and are usually en_ZA 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. dc.description.abstract AFRIKAANSE OPSOMMING: Ordinaalgetalle is transfiniete veralgemenings van die telgetalle, en word gewoonlik af_ZA 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. dc.format.extent vi, 75 pages en_ZA dc.language.iso en_ZA en_ZA dc.publisher Stellenbosch : Stellenbosch University en_ZA dc.subject Alexandrov topology en_ZA dc.subject Categories (Mathematics) en_ZA dc.subject Dedikind sums en_ZA dc.subject Grothendieck categories en_ZA dc.subject Numbers, Natural en_ZA dc.subject Set theory en_ZA dc.subject Numbers, Ordinal en_ZA dc.subject Transfinite numbers en_ZA dc.subject Axiomatic set theory en_ZA dc.subject UCTD dc.title An axiomatic approach to the ordinal number system en_ZA dc.type Thesis en_ZA dc.description.version Masters en_ZA dc.rights.holder Stellenbosch University en_ZA
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