I have been trying to find an acceptable definition of the "sum" of two natural numbers. Wikipedia has come close to what I am looking for. They begin by saying:

Let n' be the successor of n, that is the number following n in the natural numbers, so 0'=1, 1'=2. Define a+0=a.

I get all the above. They then continue by saying:

Define the general sum recursively by a + (b') = (a+b )'. Hence 1+1=1+0' = (1+ 0)'=1'=2.

I see the truth in everything that they are saying in the immediately preceding. However, I fail to see how that gives us a general definition for the sum of two natural numbers. I was really expecting something more along the lines of, "c is the sum of the natural numbers a and b if and only if blah blah blah." Anyway, can someone please tell me how I interpret the above as the definition of two natural numbers?

Thanks for any input.

... doug