The Construction of the Real Numbers

Abstract

In this thesis, there are described two standard constructions of the real numbers, these are the construction of real numbers via Dedekind cuts and the construction with metric fill of the rational numbers. Rational numbers are already a linearly ordered commutative field, so we first list the axioms of a linearly ordered commutative field. Then we take a look to the Dedekind's axiom, which only applies to real numbers and distinguishes between real and rational numbers. In thesis, there are proven some of these axioms. Beside that we prove the uniqueness theorem, which says that the set that satisfies axioms in linearly ordered commutative field, is unique up to isomorphism. Because these two constructions are quite complicated, we end thesis with method, how to explain construction of the real numbers to the secondary school students.