basic constructions:
strong axioms
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A natural number is traditionally one of the numbers $1$, $2$, $3$, and so on. It is now common in many fields of mathematics to include $0$ as a natural number as well. One advantage of doing so is that a natural number can then be identified with the cardinalities of a finite set, as well as a finite ordinal number. One can distinguish these as the nonnegative integers (with $0$) and the positive integers (without $0$), 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 $0$).
The set of natural numbers is often written $N$, $\mathbf{N}$, $\mathbb{N}$, $\omega$, or $\aleph_0$. The last two notations refer to this set's structure as an ordinal number or cardinal number respectively, and they often (usually for $\aleph$) have a subscript $0$ 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). At a foundational level, it's completely irrelevant whether $0$ counts as a natural number or not; as sets (and even as natural numbers objects), the two options are equivalent, so we are really talking about the choice of rig structure (or inclusion map into the set of integers, etc).
$\mathbf{N}$ is a natural numbers object in Set; indeed, it is the original example. This consists of an initial element $0$ (or $1$ if $0$ is not used) and a successor operation $n \mapsto n + 1$ (or simply $n \mapsto n^+$; in computer science, one often writes $n+$) such that, for a set $X$, an element $a: X$, and a function $s: X \to X$, there exists a unique function $f: \mathbf{N} \to X$ such that $f_0 = a$ and $f_{n+} = s(f_n)$. This function $f$ is said to be constructed by primitive recursion. (Fancier forms of recursion are also possible.)
The basic idea is that we define the values of $f$ one by one, starting with $f_0 = a$, then $f_1 = s(a)$, $f_2 = s(s(a))$, 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.
In classical mathematics, any inhabited subset of the natural numbers possesses a minimal element. In constructive mathematics, one cannot show this:
(a Brouwerian counterexample)
If any inhabited subset of the natural numbers possesses a minimal element, then the law of excluded middle holds.
Let $\varphi$ be an arbitrary arithmetical formula. Then the subset
is inhabited. By assumption, it possesses a minimal element $n_0$. By discreteness of the natural numbers, either $n_0 = 0$ or $n_0 \gt 0$. In the first case, $\varphi$ holds. In the second case, $\neg\varphi$ 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 $X$ are in one-to-one correspondence with upper semicontinuous functions $X \to \mathbb{N}$ (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 $X$, one obtains the well-known fact that any $\mathcal{O}_X$-module locally of finite type over $X$ is locally free on a dense open subset.
Classically, any weakly decreasing sequence of natural numbers $(a_n)$ is eventually constant, i.e. admits an index $N$ such that $a_N = a_{N+1} = a_{N+2} = \cdots$. Constructively, one can only prove that there exists an index $N$ such that $a_N = a_{N+1}$ (by induction on $a_0$); the classical principle is equivalent to the limited principle of omniscience for $\mathbb{N}$.
This is relevant to constructive algebra, as this shows that formulating chain conditions needs some care.
natural number, integer, rational number, algebraic number, Gaussian number, irrational number, real number, p-adic number