Sums of Ordinal Numbers
If an ordinal number is added to two other ordinal numbers, the order of magnitude of the sum is preserved.
Suppose
and
with
and
The set
is order isomorphic to a subset of
Call this subset
hence a function
exists.
For the sets
and
with
If
we can write 
If
we can write 
or
If
we can write
Then
where
is an order isomorphism from
onto a proper subset of
of
therefore
Also, if
then
Conversely suppose ordinal numbers exist such thatand
When
then either
or
Supposethen
which is a contraction. This rule is called left cancellation. Right cancellation is not true for ordinal numbers.