Metamath Proof Explorer

Table of Contents - 15. SURREAL NUMBERS

1. Sign sequence representation and Alling's axioms
   1. Definitions and initial properties
   2. Ordering
   3. Birthday Function
   4. Density
   5. Full-Eta Property
2. Initial consequences of Alling's axioms
   1. Ordering Theorems
   2. Birthday Theorems
3. Conway cut representation
   1. Conway cuts
   2. Zero and One
   3. Cuts and Options
   4. Cofinality and coinitiality
4. Induction and recursion
   1. Induction and recursion on one variable
   2. Induction and recursion on two variables
5. Surreal arithmetic
   1. Addition
   2. Negation and Subtraction
   3. Multiplication
   4. Division
   5. Absolute value
6. Subsystems of surreals
   1. Ordinal numbers
   2. Surreal recursive sequences
   3. Natural numbers
   4. Integers
   5. Dyadic fractions
   6. Real numbers