The Hints Promised in the Postscript to Surreal Numbers

9. To prove that x ≤ x for all numbers x in the new system, first prove the auxiliary lemma

if  x < Y_{R}  and  y ≰ x  then  x < y,

by induction on d(x), for all fixed y.

19. Perhaps the simplest answer is p = ({({1}, {1})}, {−1}). Conway has proved in fact that we can solve the equation p + p = x for any given x by taking p = ({q}, {x − q}) where

q = (X_{LL} ∪ (x + x − X_{RR}) ∪ (x + X_{L} − X_{R}), ∅).

22. Considering just the members of S created before ℵ day, the unlike numbers are precisely g_{0}, g_{1}, g_{2}, g_{3}, …, where

g_{n} = ({g_{0}, …, g_{n−1}}, {g_{0}, …, g_{n−1}}) .

Moreover, the elements {g_{0}, …, g_{n−1}} form a finite field (under Conway’s addition and multiplication operations), for n = 2, 4, 16, 256, 65536, …, 2^{2^{k}}, … !