Artmann B. The concept of number: From Quaternions to Monads and Topological Fields

New York; Toronto: John Wiley & Sons, 1988. — 262 p.The complete ordered field R
The Concept of a Field: Algebraic Preliminaries
Order Relations, Completeness
Ordered Groups and Fields
Recognition of R
On Properties Equivalent to Completion
Cantor's Characterisation of (Q, ≤)
Constructions of R
Decimal Representations of the Real Numbers
Constructions of R with Decimal Sequences
Construction of R Following Dedekind
Cantor's Construction of R
Continued Fractions
Closing Remarks on Chapter
Irrational numbers
Decimal Expansions
Algebraic Numbers
Quadratic Irrational Numbers
Transcendental Numbers
Multiples of Irrational Numbers Modulo 1
The complex numbers
Constructions of C
Some Structural Properties of C
The Fundamental Properties of C
The Fundamental Theorem as an Assertion About Extension Fields of R
Quaternions
Preliminary Remarks
Embedding R and C in H
Quaternions and Vector Calculations
The Multiplicative Group of Quaternions
Quaternions and Orthogonal Mappings in R³
The Theorems of Frobenius
The Cayley Numbers (Octaves)
Relations with Geometry 1: Vector Fields on Spheres
Relations with Geometry 2: Affine Planes
Sets and numbers
Equipotent Sets
The Number Systems as Unstructured Sets (Comparison of Cardinals)
Cardinal Numbers
Zorn's Lemma as a Proof Principle
The Arithmetic of Cardinal Numbers
Vector Spaces of Infinite Dimension, and the Cauchy Functional Equation
Appendix: Hamel's Existence Proof for a Basis B of R over Q
Non-standard numbers
Preparation: The Non- Archimedean Ordered Field R(x) of Rational Functions
The Ring ΩR of Schmieden and Laugwitz
Filters and Ultrafilters
The Fields *R(I, U) as Ultraproducts
An Axiomatic Approach to Non-Standard Analysis
Pontrjagin's topological characterization of R, C and H
Topological Groups
Topological Fields
Pontrjagin’s Theorem
Appendix : Notation and terminology
Comments on the literatureList of symbols