Georg Cantor i teorija skupova

master's thesis

• Title (english): Georg Cantor and set theory
• Author: Josip Topić
• Mentor: Boris Muha (mentor)
• Committee member: Boris Muha (predsjednik povjerenstva)
• Committee member: Siniša Slijepčević (član povjerenstva)
• Committee member: Maja Resman (član povjerenstva)
• Committee member: Nenad Antonić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-09-28, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ovaj diplomski rad istražuje život i nasljedstvo Georga Cantora, značajnog matematičara čiji su doprinosi imali dubok utjecaj na teoriju skupova i koncept beskonačnosti. U uvodnom dijelu, naglašava se važnost Cantorovih dostignuća za matematičku zajednicu i razlozi za istraživanje njegovog života. Prva sekcija rada detaljno prikazuje rani život Cantora, istražujući njegovo porijeklo, obiteljsku pozadinu te rane sklonosti prema matematici koje su oblikovale njegovu buduću karijeru. Obrazovno putovanje Cantora, koje se raspravlja u drugom dijelu, bilo je ključno za formiranje temelja njegovih matematičkih interesa. Središnji dio rada usmjerava se na Cantorovu revolucionarnu teoriju skupova. Proučavaju se povijesni okviri teorije skupova prije Cantora kako bi se razumio kontekst u kojem su se njegove ideje razvijale. Zatim se analiziraju temeljni koncepti teorije skupova koje je Cantor razvio, kao što su ekvipotentnost skupova, različite vrste beskonačnosti te značajke prebrojivih i neprebrojivih skupova. Zaključna sekcija rada sažima ključne zaključke i naglašava nasljeđe koje je Cantor ostavio u matematici. Njegova teorija skupova nije samo promijenila način na koji matematičari razmišljaju o beskonačnosti, već je postavila temelje za dublje razumijevanje strukture i relacija unutar matematičkih objekata. Kroz sve dijelove rada ističe se Cantorova strast prema istraživanju i njegova upornost u suočavanju s izazovima. Njegova vizija beskonačnosti i inovativni pristupi teoriji skupova ostavili su dubok i trajan utjecaj na matematičku misao.

Abstract (english)

This thesis explores the life and legacy of Georg Cantor, a significant mathematician whose contributions deeply influenced set theory and the concept of infinity. In the introduction, the importance of Cantor’s achievements for the mathematical community is emphasized, along with the reasons for researching his life. The first section of the paper provides a detailed portrayal of Cantor’s early life, delving into his background, family context, and early inclinations towards mathematics that shaped his future career. Cantor’s educational journey, discussed in the second part, was pivotal in laying the foundation for his mathematical interests. The central part of the paper focuses on Cantor’s revolutionary set theory. Historical frameworks of set theory prior to Cantor are studied to understand the context in which his ideas developed. Then, the fundamental concepts of set theory that Cantor developed are analyzed, such as equinumerosity of sets, various types of infinity, and features of countable and uncountable sets. The concluding section of the paper summarizes key findings and highlights the legacy Cantor left in mathematics. His set theory not only changed how mathematicians think about infinity but also laid the groundwork for a deeper understanding of the structure and relationships within mathematical objects. Throughout all sections, Cantor’s passion for research and his perseverance in facing challenges are emphasized. His vision of infinity and innovative approaches to set theory have had a profound and lasting impact on mathematical thought.

• Keywords: koncept beskonačnosti; ekvipotentnost skupova; prebrojivi skupovi; neprebrojivi skupovi
• Keywords (english): concept of infinity; equivalence of sets; features of countable and uncountable sets
• Language: croatian
• URN:NBN: urn:nbn:hr:217:664327
• Study programme: Title: Financial and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar matematike (sveučilišni magistar matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-02-06 12:58:29