I was taught under the New Math, so I should know this, but I am afraid I was tricked.
Using the cardinal, it is easy to define a multiplication, as the cardinal of the cartesian product is the product of cardinals, and a substraction, as the cardinal of a subset is the cardinal of the total set minus the cardinal of the complementary.
But the obvious use of the union of sets fails to build an addition of cardinals. You need to add also the cardinal of the intersection, and then you need a definition of sum.
Is there some way to get out of this loop, or an alternate construction? The SMSG textbook for the teacher simply suggests to use nonintersecting sets and that it is right to do not tell the pupils about the issue.
Perhaps is there some canonical family of sets so that all of them are not intersecting? Say, in the way that to define max and min we can build the family of ordinal sets.