A natural number is traditionally defined as any of the numbers , , , and so on. It is now common in many fields of mathematics to include as a natural number as well. One advantage of this is that then the natural numbers can be identified with the cardinalities of finite sets, as well as the finite ordinal numbers. One can distinguish these as the nonnegative integers (with ) and the positive integers (without ), at least until somebody uses ‘positive’ in the semidefinite sense. To a set theorist, a natural number is essentially the same as an integer, so they often use the shorter word; one can also clarify with unsigned integer (but this doesn't help with ).
The set of natural numbers is often written , , , , or . The last two notations refer to this set's structure as an ordinal number or cardinal number respectively, and they often (usually for ) have a subscript allowing them to be generalised. In the foundations of mathematics, the axiom of infinity states that this actually forms a set (rather than a proper class).
has the structure of a natural numbers object in Set; indeed, it is the original example. This consists of an initial element (or if is not used) and a successor operation (or simply ). Given any other set with an element and a function , we define by primitive recursion at a unique function such that and . (Fancier forms of recursion are also possible.) The basic idea is that we define the values of one by one, starting with , then , , and so on. These are all both possible and necessary individually, but something must be put in the foundations to ensure that this can go on uniquely forever.
(a Brouwerian counterexample)
If any inhabited subset of the natural numbers possesses a minimal element, the law of excluded middle holds.
Let be an arbitrary formula. Then the subset
is inhabited. By assumption, it possesses a minimal element . By discreteness of the natural numbers, or . In the first case, holds. In the second case, holds.
In this sense, the natural numbers are not complete, and it’s fruitful to study their completion: For instance, the global sections of the completed natural numbers object in the sheaf topos on a topological space are in one-to-one correspondence with upper semicontinuous functions (details at one-sided real number).
We can salvage the minimum principle in two ways:
Any detachable inhabited subset of the natural numbers possesses a minimal element.
Any inhabited subset of the natural numbers does not not possess a minimal element.
For instance, any finitely generated vector space over a residue field does not not possess a finite basis (pick a minimal generating set, guaranteed to not not exist). Interpreting this in the internal language of the sheaf topos of a reduced scheme , one obtains the well-known fact that any -module locally of finite type over is locally free on a dense open subset.