## Sums of Ordinal Numbers

If an ordinal number is added to two other ordinal numbers, the order of magnitude of the sum is preserved.

Supposeandwith and

The setis order isomorphic to a subset of

Call this subsethence a function

exists.

For the setsand

with

Ifwe can write

Ifwe can write or

Ifwe can write

Then

where

is an order isomorphism fromonto a proper subset ofoftherefore

Also, ifthen

Conversely suppose ordinal numbers exist such thatand

Whenthen eitheror

Supposethenwhich is a contraction. This rule is called left cancellation. Right cancellation is not true for ordinal numbers.