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

with

Bothandhave the same ordinal number,

Consider the setsandso that

so 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.