Natural numbers are the numbers used for counting, ordering, and finite repetition. They are the first numerical structure most readers meet, but their role in mathematics is not merely elementary: they provide the indexing set for [sequences](/page/Sequence), the finite sizes measured by cardinality, the exponents in powers, the iteration counts in algorithms, and the basic objects of number theory. Once the natural numbers are isolated as a structure, familiar procedures such as induction and recursive definition become theorems about that structure rather than informal habits of calculation.

Androma follows the convention that the natural numbers start at $1$:

\begin{align*} \mathbb{N} = \{1,2,3,\ldots\}. \end{align*}

The number $0$ is not a natural number under this convention, although many nearby constructions use $\mathbb{N}_0 := \mathbb{N} \cup \{0\}$ when a zero-based indexing set is needed.

## Definition

A counting system needs more than a list of numerals. It needs a first object, a rule for passing to the next object, and axioms that prevent cycles or unexpected extra elements. The Peano viewpoint packages these requirements into a structure from which arithmetic, order, and induction can be recovered.

[definition: Natural Number] The set of natural numbers is the set $\mathbb{N}$ whose elements are generated from a distinguished element $1$ by repeated application of a successor operation $S: \mathbb{N} \to \mathbb{N}$, subject to the Peano axioms: $1 \in \mathbb{N}$; $S(n) \in \mathbb{N}$ for every $n \in \mathbb{N}$; $S(n) \ne 1$ for every $n \in \mathbb{N}$; $S(m)=S(n) \implies m=n$ for all $m,n \in \mathbb{N}$; and if $A \subset \mathbb{N}$, $1 \in A$, and $n \in A \implies S(n) \in A$, then $A=\mathbb{N}$. [/definition]

The last axiom is the structural source of mathematical induction. It says that no proper subset can contain $1$ and be closed under moving to the next number. Without this condition, a system could contain a genuine copy of the counting numbers together with unrelated extra elements, and induction would not control the whole structure.

To define arithmetic from counting rather than assume it in advance, we need a primitive operation expressing only the next step. This operation is the local motion inside the natural numbers: it records how counting advances before addition, multiplication, or order have been built.

[definition: Successor Operation] A successor operation on $\mathbb{N}$ is the function $S: \mathbb{N} \to \mathbb{N}$ satisfying the Peano successor axioms $S(n) \ne 1$ for all $n \in \mathbb{N}$ and $S(m)=S(n) \implies m=n$ for all $m,n \in \mathbb{N}$. [/definition]

In ordinary decimal notation, $S(n)$ is written $n+1$. That notation already presupposes addition, so the successor map is conceptually prior to addition in an axiomatic construction.

Mathematicians often want to compare different constructions of the counting numbers. A model built from finite von Neumann ordinals, a model built from strokes on a page, and a model built abstractly from axioms should count in the same way. The following definition isolates the shared structure so that the underlying set is not mistaken for the mathematical essence.

[definition: Peano System] A Peano system is a triple $(N,e,S)$ consisting of a set $N$, an element $e \in N$, and a function $S: N \to N$ such that $S(n) \ne e$ for every $n \in N$; $S(m)=S(n) \implies m=n$ for all $m,n \in N$; and if $A \subset N$, $e \in A$, and $n \in A \implies S(n) \in A$, then $A=N$. [/definition]

The most direct model is the counting model itself. It is worth spelling out here because it shows how the abstract successor axioms correspond to everyday numerals before any later arithmetic laws are invoked.

[example: Counting Model] Take $N=\{1,2,3,\ldots\}$ and define $S:N\to N$ by $S(n)=n+1$. If $n\in N$, then $n\ge 1$, so $S(n)=n+1\ge 2$; hence $S(n)\in N$ and $S(n)\ne 1$. If $S(m)=S(n)$, then \begin{align*} m+1=n+1. \end{align*} Cancelling $1$ from both sides in the ordinary arithmetic of positive integers gives \begin{align*} m=n. \end{align*} Thus the successor map is injective. Now let $A\subset N$ with $1\in A$ and with the closure property $n\in A\implies S(n)\in A$. Since $1\in A$, closure gives \begin{align*} S(1)=1+1=2\in A. \end{align*} Applying closure again gives \begin{align*} S(2)=2+1=3\in A. \end{align*} Continuing in the same finite way, after $k-1$ successor steps from $1$ we obtain $k\in A$ for each $k\in\{1,2,3,\ldots\}$. Therefore every element of $N$ lies in $A$, so $A=N$. This concrete model satisfies the Peano axioms, so the axioms capture ordinary counting. [/example]

The axioms are also designed to reject structures that look partly like counting but contain cycles. Cycles destroy the idea of always moving to a genuinely new next number.

[example: A Cyclic Failure] Let $N=\{a,b,c\}$, choose $e=a$, and define $S:N\to N$ by $S(a)=b$, $S(b)=c$, and $S(c)=a$. For a Peano system, the successor operation must satisfy $S(n)\ne e$ for every $n\in N$. Taking $n=c$, the definition gives \begin{align*} S(c)=a. \end{align*} Since $e=a$, this is \begin{align*} S(c)=e. \end{align*} Thus the required condition $S(c)\ne e$ is false, so $(N,e,S)$ is not a Peano system. The failure is exactly cyclic behavior: starting at $e=a$, the successive values are $a$, then $S(a)=b$, then $S(b)=c$, then $S(c)=a=e$. After three successor steps the process returns to the starting element instead of producing a new next element, so this structure cannot model ordinary natural-number counting. [/example]

Another failure is subtler. The successor rule may behave correctly on a copy of the natural numbers while leaving extra elements unreachable from the starting point. The induction axiom is exactly what rules this out.

[example: Extra Unreachable Element] Let $N=\mathbb{N}\cup\{\omega\}$ with $\omega\notin\mathbb{N}$, let $e=1$, define $S(n)=n+1$ for $n\in\mathbb{N}$, and define $S(\omega)=\omega$. To test the induction axiom, take the subset $A=\mathbb{N}\subset N$. Since $e=1$ and $1\in\mathbb{N}$, we have $e\in A$. Now let $x\in A$. Then $x\in\mathbb{N}$, so the definition of $S$ on elements of $\mathbb{N}$ gives \begin{align*} S(x)=x+1. \end{align*} Because $x+1\in\mathbb{N}$ for every $x\in\mathbb{N}$, this gives \begin{align*} S(x)\in\mathbb{N}=A. \end{align*} Thus $A$ contains $e$ and is closed under $S$. However, $\omega\in N$ by the definition $N=\mathbb{N}\cup\{\omega\}$, while $\omega\notin A$ because $A=\mathbb{N}$ and $\omega\notin\mathbb{N}$. Hence \begin{align*} A\ne N. \end{align*} So there exists a proper subset of $N$ that contains $e$ and is closed under $S$, which is exactly the failure of the induction axiom in the definition of a Peano system. The extra element is unreachable from the starting point: starting at $1$, every successor step stays inside $\mathbb{N}$ because $S(n)=n+1\in\mathbb{N}$ whenever $n\in\mathbb{N}$. Since $\omega\notin\mathbb{N}$, no finite chain of successor steps beginning at $1$ can produce $\omega$. [/example]

The next structure needed for counting collections is addition: starting with $n$, count forward $m$ more steps. Defining addition by recursion prevents circularity, because it uses only the already-given successor operation and turns repeated counting into a binary operation.