Peano Axioms

You have been counting since you were very small. But have you ever stopped to wonder: what are numbers, really? The Peano axioms are mathematicians’ answer — five compact rules that conjure the entire set of counting numbers out of nothing but logic.

What is an axiom?

Mathematics is built like a tower. At the very bottom you need a foundation — facts you accept as true without proof. These bedrock truths are called axioms.

Everything above them (formulas, theorems, all the rules you learned in school) is derived by reasoning carefully from those starting points. An axiom is not proven; it is chosen. Mathematicians pick axioms that feel obviously true and are powerful enough to let them prove everything else.

Think of axioms as the rules of a board game. Before any piece moves, every player at the table agrees to the same rules. The game itself — all the strategies, moves, and outcomes — follows from those rules. In the same way, all of arithmetic follows from the Peano axioms.

What are the natural numbers?

The numbers $0, 1, 2, 3, \ldots$ are called the natural numbers. (Some older textbooks start at $1$; modern convention includes $0$, and that is what we will do here.)

The set of all natural numbers is written $\mathbb{N}$:

$$\mathbb{N} = \{0,\, 1,\, 2,\, 3,\, 4,\, \ldots\}$$

In 1889, the Italian mathematician Giuseppe Peano asked: what is the minimum you need to pin down exactly what these numbers are — no more, no less? His answer was five axioms.

The “next” button: the successor function

The central idea in Peano’s system is the successor function, written $S$. The successor of a number is simply the number right after it:

$$S(0) = 1, \qquad S(1) = 2, \qquad S(2) = 3, \qquad \ldots$$

Think of $S$ as a “next” button. Press it once starting from $0$ and you reach $1$. Press it again and you reach $2$. Because you can always press it one more time, the counting numbers never end.

Using $S$, every natural number greater than $0$ is $0$ with the button pressed some number of times:

$$1 = S(0), \qquad 2 = S(S(0)), \qquad 3 = S(S(S(0))), \qquad \ldots$$

With this picture in mind, the five axioms are easy to state and easy to understand.

The five Peano axioms

Axiom 1: zero exists

$$0 \in \mathbb{N} \tag{P1}$$

Zero is a natural number. Without a starting point the “next” button has nowhere to begin, so we declare $0$ exists. Everything else is built on top of this single seed.

Axiom 2: every number has a successor

$$\forall\, n \in \mathbb{N},\quad S(n) \in \mathbb{N} \tag{P2}$$

If $n$ is a natural number, then $S(n)$ — the number right after it — is also a natural number. This guarantees there is no last number. No matter how far you count, you can always go one further.

Axiom 3: zero is not a successor

$$\forall\, n \in \mathbb{N},\quad S(n) \neq 0 \tag{P3}$$

Zero has nothing before it. Every number other than $0$ is the successor of something, but $0$ itself is not the successor of anything.

Without this rule the numbers might “wrap around” like hours on a clock (where pressing “next” on $12$ brings you back to $1$). Axiom 3 rules that out: once you leave $0$ by pressing the “next” button, you never return to it.

Axiom 4: different numbers have different successors

$$\forall\, m, n \in \mathbb{N},\quad S(m) = S(n) \implies m = n \tag{P4}$$

If two numbers share the same successor, they must be the same number. In other words, the “next” button never sends two different numbers to the same place — each number has its own unique successor.

Imagine an endless queue of people, each standing directly behind exactly one other person. Axiom 4 says no two people ever stand behind the same person. The queue never merges; it just stretches on forever.

Axiom 5: the induction axiom

$$\Bigl[P(0)\ \land\ \forall\, n \in \mathbb{N}\,\bigl(P(n) \implies P(S(n))\bigr)\Bigr] \implies \forall\, n \in \mathbb{N},\; P(n) \tag{P5}$$

This is the most powerful — and the most surprising — of the five axioms.

In plain English: suppose some property $P$ holds for $0$, and whenever it holds for a number $n$ it also holds for the next number $S(n)$. Then $P$ holds for every natural number.

The classic image is dominoes. Picture an infinite row of standing dominoes:

• The first domino falls — that is $P(0)$.
• Each fallen domino knocks over the next one — that is $P(n) \implies P(S(n))$.
• Conclusion: every domino eventually falls — that is $\forall\, n,\; P(n)$.

Why is this axiom necessary? Without it, there could be “stray” numbers that live inside $\mathbb{N}$ but are never reached by starting at $0$ and pressing the “next” button. Axiom 5 closes the door on those impostors. It says $\mathbb{N}$ contains exactly the numbers you can reach by applying $S$ to $0$ finitely many times — nothing hidden, nothing extra.

What you can build from five axioms

Starting from (P1)–(P5) alone, you can define addition and multiplication and then prove every arithmetic rule you have ever used:

Addition is defined in two steps:

$$m + 0 \coloneqq m \tag{A1}$$ $$m + S(n) \coloneqq S(m + n) \tag{A2}$$

Rule (A1) says adding zero changes nothing. Rule (A2) says adding the successor of $n$ is the same as taking the successor of $m + n$. That is all — addition is fully captured by two lines.

Multiplication is similarly recursive:

$$m \times 0 \coloneqq 0 \tag{M1}$$ $$m \times S(n) \coloneqq (m \times n) + m \tag{M2}$$

From these definitions, and the induction axiom, you can prove:

• $m + n = n + m$ (commutativity of addition)
• $(m + n) + k = m + (n + k)$ (associativity)
• $m \times (n + k) = m \times n + m \times k$ (distributivity)
• … and everything else you know about whole-number arithmetic.

All of it is a logical consequence of five simple rules.

Summary

• An axiom is a starting assumption that is accepted without proof. All mathematical reasoning in a theory is built on top of its axioms.
• The natural numbers $\mathbb{N} = \{0, 1, 2, 3, \ldots\}$ can be defined using just five axioms, first published by Giuseppe Peano in 1889.
• The key tool is the successor function $S$: applying it once moves you to the next number.
• The five Peano axioms state:
  • (P1) $0$ is a natural number.
  • (P2) Every natural number has a successor, which is also a natural number.
  • (P3) $0$ is not the successor of any natural number — the sequence has a true beginning.
  • (P4) The successor function is injective: two different numbers always have different successors.
  • (P5) The induction axiom: any property that holds for $0$ and “spreads” from each number to its successor must hold for every natural number.
• From these five rules you can define addition and multiplication, then prove all of their familiar properties.