Addition
Natural numbers

There are two popular ways to define the sum of two natural numbers a and b. If one defines natural numbers to be the 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:

Here, A ∪ B is the union of A and B. An alternate version of this definition allows A and B 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:

Again, there are minor variations upon this definition in the literature. Taken literally, the above definition is an application of the recursion theorem on the partially ordered set N. 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 a to be temporarily "fixed", applies recursion on b to define a function " a +", and pastes these unary operations for all a together to form the full binary operation.

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.
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