Evanalysis
3.1Source-backedEstimated reading time: 1 min

3.1 Natural numbers and Peano's axioms

Move from the informal counting picture to a formal description of the natural numbers through zero, successor, and induction.

Note collections

MATH1090: Set theory

Rigorous course notes on logic, sets, and number construction, written in short linked sections with careful proofs and examples.

Language

ENEnglish繁Traditional Chinese简Simplified Chinese
All note collectionsSeries overview

Chapter 1

Logic

Reasoning tools for statements, connectives, and quantifiers.

1.1

1.1 Propositional logic

1.2

1.2 Truth tables and equivalence

1.3

1.3 Quantifiers and negation

Chapter 2

Sets and relations

Basic set language, functions, and relations.

2.1

2.1 Sets and set operations

2.2

2.2 Functions and relations

Chapter 3

Numbers by construction

How natural numbers, integers, and rationals are built, and where Q still falls short.

3.1

3.1 Natural numbers and Peano axioms

3.2

3.2 Induction and recursive arithmetic

3.3

3.3 Integers from equivalence classes

3.4

3.4 Rationals and well-defined operations

3.5

3.5 Gaps in Q and why sqrt(2) is not rational

Chapter 4

Order and completeness

Total order, bounds, supremum and infimum, and the completeness gap between Q and R.

4.1

4.1 Order, bounds, and completeness

At first sight, the natural numbers seem too familiar to need a definition. We count with them from childhood, and it is tempting to think that the list

0,1,2,3,…0, 1, 2, 3, \ldots

already says everything important.

The source notes take a more serious view. They ask what structure actually makes the natural numbers behave the way they do. The answer is not the shape of the symbols, but the presence of a distinguished starting point, a successor operation, and an induction principle.

Why a formal definition is needed

If we describe the natural numbers only by writing 0,1,2,3,…0, 1, 2, 3, \ldots, then we have not really explained what the dots mean, why the process continues, or why induction works.

The Peano viewpoint solves that problem by specifying the essential properties directly. It tells us what must be true in any model of the natural numbers, without depending on intuition alone.

The data of a model

Definition

A model of the natural numbers

Suppose NN is a set equipped with:

  • a distinguished element 0∈N0 \in N;
  • a function S:N→NS : N \to N, called the successor map.

The triple (N, 0, S) is called a model of the natural numbers if it satisfies the Peano axioms:

  1. SS is injective: if S(x)=S(y)S(x) = S(y), then x=yx = y.
  2. No element is its own successor: S(x)≠xS(x) \ne x for every x∈Nx \in N.
  3. Zero is not a successor: there is no x∈Nx \in N with S(x)=0S(x) = 0.
  4. Induction holds: if a predicate PP is true of 0, and whenever P(x) is true it follows that P(S(x)) is true, then PP is true for every x∈Nx \in N.

The central idea is that the natural numbers are characterized by how they are connected, not by how they are written.

What each axiom is doing

Each axiom rules out a specific kind of pathology.

  • Injectivity says different numbers cannot suddenly merge after one successor step.
  • S(x)≠xS(x) \ne x rules out fixed points.
  • S(x)≠0S(x) \ne 0 says zero is the starting point, not something reached later.
  • Induction rules out disconnected extra pieces and ensures that every element lies in the chain generated from 0.

Taken together, these axioms force the familiar picture of counting forward one step at a time.

Reading numbers through successor

Worked example

How the usual numerals arise from 0 and SS

Once 0 and the successor map are fixed, the next numbers are interpreted as

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

and so on.

So the notation 2 is shorthand for "the element obtained by applying the successor map twice to 0." The notation is convenient, but the structure comes first.

This is why the notes sometimes return to successor notation when they want the definition to remain visible instead of being hidden behind familiar symbols.

Induction is not an extra trick

Students often meet induction as a proof technique after they already believe the natural numbers are understood. The source notes reverse that perspective.

The induction principle is part of the definition of what the natural numbers are. In that sense, induction is not merely a useful method for proving statements about NN; it is one of the structural facts that makes NN the natural numbers in the first place.

Theorem

What induction gives you

To prove a statement P(n) for all n∈Nn \in N, it is enough to show:

  1. P(0) is true.
  2. For every x∈Nx \in N, if P(x) is true, then P(S(x)) is true.

Once these two facts are established, induction implies that P(n) holds for every natural number n.

A model that fails

Worked example

Why a finite cycle is not a model of the natural numbers

Consider the set \{0,1,2\} with successor map

S(0)=1,S(1)=2,S(2)=0.S(0)=1, \qquad S(1)=2, \qquad S(2)=0.

This structure does not satisfy the Peano axioms.

First, 0 is a successor because S(2)=0S(2)=0, so axiom 3 fails. Second, induction fails as well: the subset \{0,1,2\} loops back on itself instead of extending in a one-way chain from 0.

So although the symbols look familiar, this structure is not a model of the natural numbers.

This example is important because it shows why the Peano axioms are not ornamental. They exclude structures that resemble counting in superficial ways but do not behave like NN.

Why the axioms are more informative than the dots

The expression

0,1,2,3,…0,1,2,3,\ldots

is useful notation, but by itself it does not explain why arithmetic and induction work.

The Peano description does. It tells us:

  • where counting begins;
  • how one step forward is represented;
  • why there are no loops;
  • why proofs by induction reach every natural number.

That is why the formal definition is more revealing than the informal list.

Common mistakes

Common mistake

Do not confuse the symbol with the structural role

The Peano viewpoint does not say that the written mark 2 has some built-in meaning. It says that the object denoted by 2 is the second successor of 0.

Common mistake

Induction is not optional decoration

Without the induction axiom, a structure can contain a familiar successor chain starting from 0 and still have extra disconnected elements or loops. The induction principle rules those out.

Quick checks

Quick check

Why does the axiom S(x)≠0S(x) \ne 0 matter?

Explain what would go wrong if 0 were allowed to be a successor.

Solution

Answer

Quick check

What does injectivity of the successor map prevent?

Answer in terms of two different numbers trying to behave like the same next number.

Solution

Answer

Quick check

After proving a base case and an induction step, what exactly may you conclude?

State the conclusion carefully.

Solution

Answer

Exercises

Quick check

Show that the structure \{0,1,2\} with successor map S(0)=1S(0)=1, S(1)=2S(1)=2, S(2)=2S(2)=2 is not a model of the natural numbers. Which axiom fails?

Check the axioms one by one instead of guessing.

Solution

Guided solution

Read this first

This note sits at the start of the construction chapter. It connects forward to 3.2 Induction and recursive arithmetic and uses language prepared earlier in 2.2 Functions and relations.

Prerequisites

This section can be read on its own.

Key terms in this unit

More notes in this series

Previous section

2.2 Functions and relations

Next section

3.2 Induction and recursive arithmetic