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 ωα apparently increases much faster than α, there exist ordinals such ωα = a. Such ordinals, known as epsilon numbers, can be arranged in an increasing transfinite sequence ε0,ε1 … εα. Not only is this sequence defined for all α but there exist α such that εa = α. Furthermore, this construction can be extended indefinitely. Specifically, let us use the notation ε0 (α) = εa and let ε 1(0) be the least α such that ε0(α) = a. Then for every ordinal β, there is an increasing sequence εb(0) εb(1), … εβ(ω) where εβ (a) runs through all ordinals which are fixed for every εγ with γ > β.