Chapter 3 - The Basic Operations

Summary

ADDITION

We define addition by induction on the natural sum of the lengths of the addends. Recall that the natural sum is obtained by expressing the ordinals in normal form in terms of sums of powers of ω and then using ordinary polynomial addition, in contrast to ordinary ordinal addition which has absorption. Thus the natural sum is a strictly increasing function of each addend.

The following notation will be convenient. If a = F|G is the canonical representation of a, then a′ is a typical element of F and a″ is a typical element of G. Hence a′ < a < a″. We are now ready to give the definition.

Definition. a + b = {a′+b, a+b′}|{a″+b, a+b″}.

Several remarks are appropriate here. First, since the induction is on the natural sum of the lengths, we are permitted to use sums such as a′+b in the definition. Secondly, no further definition is needed for the beginning. Since at the beginning we have only the empty set, we can use the trite remark that {f(x):xεφ} = φ regardless of f. For example, φ|φ + φ|φ = φ|φ. Thirdly, there is the a priori possibility that the sets F and G used in the definition of a+b do not satisfy F < G. To make the definition formally precise, one can use the convention that F|G = u for some special symbol u if F < G and that F|G = u if u ε F∪G.