Chapter 6 - Lengths and Subsystems which are Sets

Summary

Up to now we have considered basic material which in some form is contained in [1], At this point we begin to consider problems that are new and more specialized. In this chapter we are interested in information regarding upper bounds of lengths of surreal numbers obtained by various operations. This will allow us to obtain subclasses of the proper class of surreal numbers which are actually subsets and closed under desirable operations. Our first result is an easy one.

Theorem 6.1. l(a+b) ≤ l(a) ⊕ l(b).

Proof. We use induction as usual. A typical upper or lower element in the canonical representation of a+b has the form a⁰+b or a+b⁰. Without loss of generality consider a⁰+b. By the inductive hypothesis l(a⁰+b) ≤ l(a⁰+b) ⊕ ≤ l(b) < l(a) ⊕ l(b). The result follows by theorem 2.3.

This result is sharp. In fact, we already know that in the special case where a and b are ordinals we obtain equality. If a and b are not ordinals then we usually have a proper inequality. For example, l(1) = 1 but l(½) = 2, hence l(1) ≤ l(½) + l(½).

Incidentally, the sign sequence formula of the preceding chapter makes it comparatively routine to study the question of when equality is obtained. We do not pursue this here since we are at present more interested in bounds.