constructing the reals
The real numbers can be explicitly constructed as sets of rational numbers using the Dedekind cut construction.
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.)
- 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)
→ 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
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