Editing Natural number (section)

===Addition===
Given the set <math>\mathbb{N}</math> of natural numbers and the [[successor function]] <math>S \colon \mathbb{N} \to \mathbb{N}</math> sending each natural number to the next one, one can define [[Addition in N|addition]] of natural numbers recursively by setting {{math|''a'' + 0 {{=}} ''a''}} and {{math|''a'' + ''S''(''b'') {{=}} ''S''(''a'' + ''b'')}} for all {{math|''a''}}, {{math|''b''}}. Thus, {{math|''a'' + 1 {{=}} ''a'' + S(0) {{=}} S(''a''+0) {{=}} S(''a'')}}, {{math|''a'' + 2 {{=}} ''a'' + S(1) {{=}} S(''a''+1) {{=}} S(S(''a''))}}, and so on. The [[algebraic structure]] <math>(\mathbb{N}, +)</math> is a [[commutative]] [[monoid]] with [[identity element]]&nbsp;0. It is a [[free object|free monoid]] on one generator. This commutative monoid satisfies the [[cancellation property]], so it can be embedded in a [[group (mathematics)|group]]. The smallest group containing the natural numbers is the [[integer]]s.

If 1 is defined as {{math|''S''(0)}}, then {{math|''b'' + 1 {{=}} ''b'' + ''S''(0) {{=}} ''S''(''b'' + 0) {{=}} ''S''(''b'')}}. That is, {{math|''b'' + 1}} is simply the successor of {{math|''b''}}.