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.