Proof That Changing Order of Union of Sets Changes Ordinal Number
Supposeandwith bothandhaving the natural order.
B is isomorphic under to and onto the setwith
Bothandhave the same ordinal number,
Consider the setsandso that
so the order of union of two sets does matter with respect to ordinal numbers.
We can also writeso it is also true that
This means that ordinal numbers are not commutative under addition.