Proof That Changing Order of Union of Sets Changes Ordinal Number
Suppose
and
with both
and
having the natural order.
B is isomorphic under to and onto the set
with
with
Bothand
have the same ordinal number,
Consider the sets
and
so that
so that
so the order of union of two sets does matter with respect to ordinal numbers.
We can also write
so it is also true that
This means that ordinal numbers are not commutative under addition.
