Editing Addition (section)

=== Natural numbers ===
{{Further|Natural number}}
There are two popular ways to define the sum of two natural numbers <math> a </math> and <math> b </math>. If one defines natural numbers to be the [[Cardinal number|cardinalities]] of finite sets (the cardinality of a set is the number of elements in the set), then it is appropriate to define their sum as follows:{{sfnmp
 | 1a1 = Begle | 1y = 1975 | 1p = 49
 | 2a1 = Johnson | 2y = 1975 | 2p = 120
 | 3a1 = Devine | 3a2 = Olson | 3a3 = Olson | 3y = 1991 | 3p = 75
}}
{{blockquote|Let <math> N(S) </math> be the cardinality of a set <math> S </math>. Take two disjoint sets <math> A </math> and <math> B </math>, with <math> N(A) = a </math> and <math> N(B) = b </math>. Then <math> a + b </math> is defined as <math> N(A \cup B)</math>.
}}
Here <math> A \cup B </math> means the [[union (set theory)|union]] of <math> A </math> and <math> B </math>. An alternate version of this definition allows <math> A </math> and <math> B </math> to possibly overlap and then takes their [[disjoint union]], a mechanism that allows common elements to be separated out and therefore counted twice.

The other popular definition is recursive:{{sfnp|Enderton|1977|p=[http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA79 79]}}
{{blockquote|Let <math> n^+ </math> be the successor of <math> n </math>, that is the number following <math> n </math> in the natural numbers, so <math> 0^+ = 1 </math>, <math> 1^+ = 2 </math>. Define <math> a + 0 = a </math>. Define the general sum recursively by <math> a + b^+ = (a + b)^+ </math>. Hence <math> 1 + 1 = 1 + 0^+ = (1 + 0)^+ = 1^+ = 2 </math>.
}}
Again, there are minor variations upon this definition in the literature. Taken literally, the above definition is an application of the [[Recursion#The recursion theorem|recursion theorem]] on the [[partially ordered set]] <math> \mathbb{N}^2 </math>.<ref>For a version that applies to any poset with the [[descending chain condition]], see {{harvtxt|Bergman|2005}}, p. 100</ref> On the other hand, some sources prefer to use a restricted recursion theorem that applies only to the set of natural numbers. One then considers <math> a </math> to be temporarily "fixed", applies recursion on <math> b </math> to define a function "<math> a + </math>", and pastes these unary operations for all <math> a </math> together to form the full binary operation.<ref>{{harvtxt|Enderton|1977}}, p. [http://books.google.com/books?id=JlR-Ehk35XkC&pg=PA79 79] observes, "But we want one binary operation <math> + </math>, not all these little one-place functions."</ref>

This recursive formulation of addition was developed by Dedekind as early as 1854, and he would expand upon it in the following decades. He proved the associative and commutative properties, among others, through [[mathematical induction]].{{sfnp|Ferreirós|1999|p=223}}