# constructing the reals

## Summary

The real numbers can be explicitly constructed as sets of rational numbers using the Dedekind cut construction.

## Context

This concept has the prerequisites:

- real numbers
- set operations (The Dedekind cut construction defines real numbers as sets.)
- order relations (One must define an order relation on the reals.)
- fields (One must show that the construction satisfies the field axioms.)

## Goals

- Define the real numbers using the Dedekind cut construction.

- Show that this construction satisfies the least upper bound property.

- Define the basic arithmetic operations and comparison operators

- Show that these satisfy the field axioms

## Core resources (read/watch one of the following)

## -Paid-

→ Principles of Mathematical Analysis

A classical undergraduate analysis textbook.

Location:
Appendix to Chapter 1, "The real and complex number systems," pages 17-21

→ Elements of Set Theory

An introductory textbook on axiomatic set theory.

Location:
Chapter 5, "Construction of the real numbers," subsection "Real numbers," pages 111-120

## See also

-No Additional Notes-