Natural Numbers and Axioms

Entry conditions

Use axiomatic arithmetic only when you need a formal foundation for numbers and their operations, not just an informal counting story.

Definitions

The Peano axioms describe the natural numbers $(\mathbb{N},0,S)$:

1. $0$ is a natural number.
2. If $n$ is a natural number, then $S(n)$ is a natural number.
3. $0$ is not the successor of any natural number.
4. If $S(n)=S(m)$, then $n=m$.
5. Induction: if a set contains $0$ and is closed under successor, it contains all natural numbers.

Vocabulary (plain language)

• Axiom: a rule accepted as a starting point.
• Successor: the operation that produces the next number.
• Induction: a proof rule for all natural numbers.

Symbols used

• $\mathbb{N}$: the natural numbers.
• $S(n)$: the successor of $n$.

Intuition

Rather than saying “numbers are what you count,” the axioms say: start with $0$ and repeatedly apply successor. This gives a structure where “counting” can be defined and proved.

Worked example

Using the axioms, $1$ is defined as $S(0)$, $2$ as $S(1)$, and so on. These are not assumptions; they are constructed by repeated successor.

How to recognize the structure

• You have a base element ($0$).
• You have a successor operation.
• You can apply induction to prove statements for all natural numbers.

Common mistakes

• Treating “natural numbers” as obvious without specifying axioms.
• Using induction without proving the base case.