Ordinal Arithmetic - Addition

Addition

The union of two disjoint well-ordered sets S and T can be well-ordered. The order-type of that union is the ordinal which results from adding the order-types of S and T. If two well-ordered sets are not already disjoint, then they can be replaced by order-isomorphic disjoint sets, e.g. replace S by S × {0} and T by T × {1}. Thus the well-ordered set S is written "to the left" of the well-ordered set T, meaning one defines an order on S T in which every element of S is smaller than every element of T. The sets S and T themselves keep the ordering they already have. This addition is associative and generalizes the addition of natural numbers.

The first transfinite ordinal is ω, the set of all natural numbers. Let's try to visualize the ordinal ω + ω: two copies of the natural numbers ordered in the normal fashion and the second copy completely to the right of the first. If we write the second copy as {0' < 1' < 2', ...} then ω + ω looks like

0 < 1 < 2 < 3 < ... < 0' < 1' < 2' < ...

This is different from ω because in ω only 0 does not have a direct predecessor while in ω + ω the two elements 0 and 0' do not have direct predecessors. Here are 3 + ω and ω + 3:

0 < 1 < 2 < 0' < 1' < 2' < ...

0 < 1 < 2 < ... < 0' < 1' < 2'

After relabeling, the former just looks like ω itself while the latter does not: we have 3 + ω = ω. But ω + 3 is not equal to ω since ω + 3 has a largest element (namely, 2') and ω does not. So our addition is not commutative.

However, addition is still associative; one can see for example that (ω + 4) + ω = ω + (4 + ω) = ω + ω.

The definition of addition can also be given inductively (the following induction is on β):

• α + 0 = α,
• α + (β + 1) = (α + β) + 1 (here, "+ 1" denotes the successor of an ordinal),
• and if δ is a limit ordinal then α + δ is the limit of the α + β for all β < δ.

Using this definition, we also see that ω + 3 is a successor ordinal (it is the successor of ω + 2) whereas 3 + ω is the limit of 3 + 0 = 3, 3 + 1 = 4, 3 + 2 = 5, etc., which is just ω.

Zero is an additive identity α + 0 = 0 + α = α.

Addition is associative (α + β) + γ = α + (β + γ).

Addition is strictly increasing and continuous in the right argument:

but the analogous relation does not hold for the left argument; instead we only have:

Ordinal addition is left-cancellative: if α + β = α + γ, then β = γ. Furthemore, one can define left subtraction for ordinals β ≤ α: there is a unique γ such that α = β + γ. On the other hand, right cancellation does not work:

but

Nor does right subtraction, even when β ≤ α: for example, there does not exist any γ such that γ + 42 = ω.