# Michal Mnich: Real numbers - Cantor’s approach and infinite decimal expansions

## Bachelor thesis presentation

**Time: **
Tue 2022-06-07 09.00 - 10.00

**Location: **
Kräftriket, house 5, room 32

**Respondent: **
Michal Mnich

**Abstract:**

This paper presents two different constructions of the real numbers based on the existence of the field of rational numbers. The concepts of supremum, total orderings, fields and isomorphisms are introduced with basic set theory as a starting point. An intuitive approach to real numbers through infinite decimal expansions is presented and shown to result in a totally ordered field with the supremum property. A different approach by means of Cauchy sequences is also presented and shown to result in a totally ordered field with the supremum property. Finally, it is shown that any two totally ordered fields with the supremum property are isomorphic, and the real numbers are defined as any such field. It is argued that both presented constructions of the real numbers have their uses in different circumstances.