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An axiomatic approach to the ordinal number system
Thesis (MSc)--Stellenbosch University, 2021.
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| Format: | Thesis |
| Language: | en_ZA |
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Stellenbosch : Stellenbosch University
2021
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| _version_ | 1867613941406892032 |
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| access_status_str | Open Access |
| author | Van der Berg, Ineke |
| author2 | Janelidze, Zurab |
| author_browse | Janelidze, Zurab Van der Berg, Ineke |
| author_facet | Janelidze, Zurab Van der Berg, Ineke |
| author_sort | Van der Berg, Ineke |
| collection | Thesis |
| dc_rights_str_mv | Stellenbosch University |
| description | Thesis (MSc)--Stellenbosch University, 2021. |
| format | Thesis |
| id | oai:scholar.sun.ac.za:10019.1/109901 |
| institution | Stellenbosch University (South Africa) |
| language | en_ZA |
| last_indexed | 2026-06-10T12:44:08.546Z |
| license_str | Other — see source repository |
| provenance_str_mv | Harvested via OAI-PMH from SUNScholar — Stellenbosch University Repository |
| publishDate | 2021 |
| publishDateRange | 2021 |
| publishDateSort | 2021 |
| publisher | Stellenbosch : Stellenbosch University |
| publisherStr | Stellenbosch : Stellenbosch University |
| record_format | dspace |
| source_str | SUNScholar — Stellenbosch University Repository |
| spelling | oai:scholar.sun.ac.za:10019.1/109901 An axiomatic approach to the ordinal number system Van der Berg, Ineke Janelidze, Zurab Stellenbosch University. Faculty of Science. Dept. of Mathematical Sciences. Division Mathematics. Alexandrov topology Categories (Mathematics) Dedikind sums Grothendieck categories Numbers, Natural Set theory Numbers, Ordinal Transfinite numbers Axiomatic set theory UCTD Thesis (MSc)--Stellenbosch University, 2021. 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. Masters 2021-03-06T12:36:30Z 2021-04-21T14:31:07Z 2021-03-06T12:36:30Z 2021-04-21T14:31:07Z 2021-03 Thesis http://hdl.handle.net/10019.1/109901 en_ZA Stellenbosch University vi, 75 pages application/pdf Stellenbosch : Stellenbosch University |
| spellingShingle | Alexandrov topology Categories (Mathematics) Dedikind sums Grothendieck categories Numbers, Natural Set theory Numbers, Ordinal Transfinite numbers Axiomatic set theory UCTD Van der Berg, Ineke An axiomatic approach to the ordinal number system |
| title | An axiomatic approach to the ordinal number system |
| title_full | An axiomatic approach to the ordinal number system |
| title_fullStr | An axiomatic approach to the ordinal number system |
| title_full_unstemmed | An axiomatic approach to the ordinal number system |
| title_short | An axiomatic approach to the ordinal number system |
| title_sort | axiomatic approach to the ordinal number system |
| topic | Alexandrov topology Categories (Mathematics) Dedikind sums Grothendieck categories Numbers, Natural Set theory Numbers, Ordinal Transfinite numbers Axiomatic set theory UCTD |
| url | http://hdl.handle.net/10019.1/109901 |
| work_keys_str_mv | AT vanderbergineke anaxiomaticapproachtotheordinalnumbersystem AT vanderbergineke axiomaticapproachtotheordinalnumbersystem |