Are the definitions of addition of natural numbers as cardinals ($|A| + |B| =|A \cup B|$) and by recursion ($a+S(b) = S(a+b)$) related?

Question

In Peterson, Theory of Arithmetic ( available at Archive.org) addition of whole numbers is defined in the following way.

If $A$ and $B$ are disjoint sets, and if $a$ is the cardinal number of $A$ and $b$ is the cardinal number of $B$, then $a + b$ is the cardinal number of $A\cup B$.

In set theory, addition of natural numbers is defined in the following way:

1. $a+0=a$
2. $a+S(b) = S(a+b)$.

How do these two definitions relate to one another? Should one consider that two different operations are defined? Or are we dealing here with two different ways to define the same operation?

Answer 1

In set theory the natural numbers play two different roles:

1. Cardinals.
2. Ordinals.

This is because of the peculiar property: there is essentially a unique way to well-order (or indeed linearly order) a finite set. And this becomes flagrantly false for infinite sets (at least those which can be well-ordered).

The arithmetic of cardinals is very incompatible with the arithmetic of ordinals, except for the finite case. For example, ordinal arithmetic is not commutative whereas cardinal arithmetic is.

This means that we can define the arithmetic of the natural numbers in two ways: treat them as cardinals, that is the first way you present; or treat them as ordinals, that is the second way you present.

Both methods are equivalent, and we can prove that assuming one of them holds, that the second holds as well. So this is really about what is convenient to you at the moment. It makes conceptual sense, especially if you are to talk about induction, to define the addition in the ordinal sense. But at the same time, you might want to have a definition which is induction free.

Of course, once you move away from set theory, the natural numbers tend to not be sets anymore, and then going through cardinal arithmetic becomes very convoluted and unnecessary (not to mention that in some of these contexts addition is already given). And on the other hand, the well-foundedness of the natural numbers (or equivalently: the susceptibility of the natural numbers to induction-based arguments) is an essential property of them, making the ordinal definition far easier to implement.

Answer 2

One wants cardinality to be a measure on sets and, for finite sets, the result of the measurement to be a natural number. For that, we can either define the measure of the disjoint union to obey the rules of addition of natural numbers, or define addition of natural numbers to obey the rules that make cardinality a measure.