LMS: 110 Introduction Theory of Surreal Numb (London Mathematical Society Lecture Note Series, Series Number 110)

The surreal numbers form a system which includes both the ordinary real numbers and the ordinals. Since their introduction by J. H. Conway, the theory of surreal numbers has seen a rapid development revealing many natural and exciting properties. These notes provide a formal introduction to the theory in a clear and lucid style. The the author is able to lead the reader through to some of the problems in the field. The topics covered include exponentiation and generalized e-numbers. Cover London Mathematical Society Lecture Note Series 110 An Introduction to the Theory of Surreal Numbers Copyright 9780521312059 Contents Preface Chapter 1 Introduction Chapter 2 Definition and Fundamental Existence Theorem A. Definition B. Fundamental Existence Theorem C. Order Properties Chapter 3 The Basic Operations A. Addition B. Multiplication C. Division D. Square Root Chapter 4 Real Numbers and Ordinals A. Integers B. Dyadic Fractions C. Real Numbers D. Ordinals Chapter 5 Normal Form A. Combinatorial Lemma on Semigroups C. Normal Form D. Application to Real Closure E. Sign Sequence Chapter 6 Lengths and Subsystems which are Sets Chapter 7 Sums as Subshuffles, Unsolved Problems Chapter 8 Number Theory A. Basic Results B. Partial Results and Unsolved Problems Chapter 9 Generalized Epsilon Numbers A. Epsilon Numbers with Arbitrary Index B. Higher Order Fixed Points C. Sign Sequences for Fixed Points E. Sign Sequences in Quasi Case Chapter 10 Exponentiation A. General Theory B. Specialization to Purely Infinite Numbers C. Reduction to the Function g D. Properties of g and Explicit Results References Index These notes provide a formal introduction to the theory of surreal numbers in a clear and lucid style