Download Mathematics 317

Mathematics 317
Foundations of Analysis
Text: Number Systems, by Elliott Mendelson
Chapter 2: The Natural Numbers
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
Peano Systems
The Iteration Theorem
Application of the Iteration Theorem: Addition
The Order Relation
Multiplication
Exponentiation
Isomorphism, Categoricity
A Basic Existence Assumption
Supplementary Exercises
Suggestions for Further Reading
Chapter 3: The Integers
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3
Rational Numbers
Fields
Quotient Field of an Integral Domain
Ordered Field
Subfields, Rational Numbers of a Field
Chapter 5: The Real Number System
5.1
5.2
5.3
3
Definition of the Integers
Addition and Multiplication of Integers
Rings and Integral Domains
Ordered Integral Domains
Greatest Common Divisor, Primes
Integers Modulo n
Characteristic of an Integral Domain
Natural Numbers and Integers of an Integral Domain
Subdomains, Isomorphisms, Characterizations of the Integers
Supplementary Exercises
Chapter 4: Rational Numbers and Ordered Fields
4.1
4.2
4.3
4.4
4.5
3 wks.
Inadequacy of the Rationals
Archimedean Ordered Fields
Least Upper Bounds and Greatest Lower Bounds
5
5.4
5.5
5.6
5.7
5.8
5.9
The Categoricity of the Theory of Complete Ordered Fields
Convergent Sequences and Cauchy Sequences
Cauchy Completion. The Real Number System
Elementary Topology of the Real Number System
Continuous Functions
Infinite Series