Chapter 9 - Generalized Epsilon Numbers

Summary

EPSILON NUMBERS WITH ARBITRARY INDEX

On page 35 in [1] Conway makes some remarks on the possibility of extending the transfinite sequence of epsilon numbers to more general indices, e.g. he gives a meaning to ε_-1. He also mentions other interesting surreal numbers. He mentions that the equation ω^-x = x has a unique solution and that there exist various pairs (x,y) satisfying ω^-x =y and ω^-x = x. In this chapter we study this systematically. Moreover, we also discuss higher order fixed points, e.g. in the sequence of ordinary ε numbers ε₀, ε₁ … there exist α such that ε _α = α and such a can be parametrized by ordinals. It is interesting that a general elegant theory exists for surreal numbers which have such fixed point properties.

First, we summarize the situation for ordinals. It comes as a surprise to the beginner that although ω^α apparently increases much faster than α, there exist ordinals such ω^α = a. Such ordinals, known as epsilon numbers, can be arranged in an increasing transfinite sequence ε₀,ε₁ … ε_α. Not only is this sequence defined for all α but there exist α such that ε_a = α. Furthermore, this construction can be extended indefinitely. Specifically, let us use the notation ε₀ (α) = ε_a and let ε ₁(0) be the least α such that ε₀(α) = a. Then for every ordinal β, there is an increasing sequence ε_b(0) ε_b(1), … ε_β(ω) where ε_β (a) runs through all ordinals which are fixed for every ε_γ with γ > β.