Number systems: an axiomatic foundation for algebra,
number theory, and analysis
This course carefully and rigorously develops several number systems
including the set of natural numbers N, the ring of integers Z, the ring
of integers modulo m, the field of rational numbers Q, the field of real numbers R,
and the field of complex numbers C. This course has been taught
for several years by Wayne Aitken and Linda Holt
as Math 378 at Cal State San Marcos.
A major purpose of this course is to illustrate the axiomatic method. As such
it will emphasize careful proof in a mostly familiar setting.
Another purpose is to develop a strong but accessible foundation for higher mathematics.
This course develops foundational ideas needed for
more advanced courses
including Abstract Algebra, Analysis, Number Theory.
This course follow a long tradition of developing the number systems axiomatically
starting with Dedekind, Cantor, and Peano.
Some notable precursors that were consulted include the following:
The Number Systems: Foundations of Algebra and Analysis,
by Solomon Feferman
Foundations of Analysis, by Edmund Landau
Prerequisites
Basic proficiency with logic, proofs, set theory,
functions (injective, surjective, bijective, inverses), and relations (order relations and equivalence relations).
At CSUSM, this precursory material is covered in Math 350, but a good discrete mathematics course
might have much of the needed material.
I recommend the following textbook in order to get prepared for this course:
How to Prove It: A Structured Approach
Daniel J. Velleman
Cambridge University Press (2nd Edition)
Summer 2019 version
This course has been evolving since 2007.
What follows is the Summer 2019 version of the course.
(The early drafts were written by Wayne Aitken. Later
versions from around 2013 on were cowritten by Wayne Aitken and Linda Holt).
Chapter 1.
Peano's Axioms. Natural Numbers N.
Chapter 2.
Order in N
Chapter 3.
Cardinality and Counting (Further exploration of N)
Chapter 4.
The construction of the integers Z.
Chapter 5.
Exploring Z.
Chapter 6.
Modular arithmetic (including the
ring of integers modulo n, the field of integers mod p).
Chapter 7.
The field Q (and ordered fields in general).
Chapter 8.
Sequences and limits in ordered fields.
Chapter 9.
The ideas of completeness and continuity for ordered fields.
Chapter 10.
The official construction of the real numbers using Cauchy sequences.
Chapter 11.
Exploring R
Chapter 12.
The Complex numbers C.
Appendix A: Chapter 0. Background: logic and set theory.
Appendix B. Further exploration
of C.
Appendix C.
Polynomials.