Hausdorffov paradoks

Brezec, Sara (author), Slapar, Marko (mentor)

V diplomskem delu obravnavamo Hausdorffov paradoks, ki pravi, da je sfera, ki ji odvzamemo končno mnogo točk, paradoksalna. Najprej obravnavamo koncept neskončne množice. Nato se osredotočimo na aksiom izbire, ki je ključen za dokazovanje Hausdorffovega paradoksa. Namen naslednjega poglavja o rotacijskih grupah je ponovitev znanja, ki je ključno za razumevanje sledečih izrekov in dokazov. V četrtem poglavju govorimo o skladnosti likov s stališča teorije množic. Preden definiramo paradoksalnost množice, ponovimo še pojem delovanja grupe. Obravnavamo tudi prosto grupo ranga 2, ki jo potrebujemo za dokazovanje paradoksa. Šesto poglavje je namenjeno paradoksom ravnine, v sedmem poglavju pa kot končen rezultat navedemo dokaz Hausdorffovega paradoksa.

• Language: Slovenian
• Keywords: Hausdorffov paradoks
• Work type: Bachelor thesis/paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: PEF - Faculty of Education
• Year: 2016
• PID: 20.500.12556/RUL-85977
• COBISS.SI-ID: 11196745
• Publication date in RUL: 19.09.2017

Secondary language

• Language: English
• Title: Hausdorff paradox

Abstract

In this diploma paper we will deal with the Hausdorff paradox, which says, that the sphere without a finite number of points is paradoxical. First we cover the concept of an infinite set. Then we focus on the axiom of choice, which is essential for prooving the Hausdorff paradox. The purpose of the next chapter on rotation groups is just a revision of knowledge, crucial for understanding further theorems and proofs. In Chapter 4 we talk about a congruence of figures based on the set theory. Before defining the paradoxical set, we revise the concept of a group action. We also deal with the free group of rank 2, which is required to prove the paradox. Chapter 6 is about paradoxes of the plane, in Chapter 7 we finally present a proof of the Hausdorff paradox.

• Keywords: Hausdorff paradox