Tao's Real Analysis - Peano Axiom

We are forever friends.
Yes?
– Rhypho L.

0. Preface

Terence Tao, in his Analysis I, introduces a remarkably classical example of an axiomatic system: the Peano axioms, which provide a rigorous foundation for the natural numbers and settle elementary questions such as why $1+1=2$ .

Natural numbers are among the first things we encounter in education, yet they are seldom formally defined or critically examined. This example offers the reader a glimpse into how a complete mathematical theory can be erected from the barest possible axiomatic scaffolding — a handful of statements accepted without proof, from which everything else follows.

1. Peano Axioms

Axiom 1.1

$0$ is a natural number.

This single statement anchors the entire system. It is natural — in every sense of the word — to begin counting from $0$ . Note that at this stage we have not yet formally defined $2$ , or any other numeral; nor is our theory tied to any particular numeral system such as decimal, binary, or octal.

We write $n++$ for the successor of a natural number $n$ , so that the sequence $0,0++,(0++)++,\dots$ is intended to exhaust all natural numbers. The next axiom makes this precise.

Axiom 1.2

If $n$ is a natural number, then $n++$ is also a natural number.

This axiom generates the full natural number system by recursion.

Axiom 1.3

$0$ is not the successor of any natural number.

Without this axiom the system could wrap around — counting $0,1,\dots ,65535,0,\dots$ as an unsigned short does in C. We rule out such cyclic behaviour by fiat.

Axiom 1.4

If $n\ne m$ , then $n++\ne m++$ .
Equivalently, distinct natural numbers have distinct successors.

Axiom 1.5: Principle of Mathematical Induction

Let $P(n)$ be any property of a natural number $n$ . If $P(0)$ is true, and if
$P(n)$ being true implies $P(n++)$ is true, then $P(n)$ holds for every
natural number $n$ .

Axiom 1.5 is the workhorse of the theory. It not only provides a proof technique but also licenses recursive definitions of operations. Crucially, it extends our conclusions to all of countably infinite $\mathbb{N}$ .

2. Addition

We now introduce addition as a binary operation on natural numbers, denoted $+$ , defined recursively with the aid of Axiom 1.5. The inductive template is:

1. Establish the base case: $P(0)$ holds.
2. Inductive hypothesis: assume $P(n)$ holds.
3. Inductive step: show $P(n++)$ is well-defined and true.

Definition 2.1

Let $m$ be a natural number. Define

$$0+m:=m.$$

If $n+m$ has already been defined, set

$$(n++)+m:=(n+m)++.$$

Example 2.1.1. $0+m=m$ , $(0++)+m=m++$ , and so on.

Remark 2.1. One may think of $(n+\cdot )$ as a unary operation that applies the successor map exactly $n$ times. The inductive definition guarantees it is well-defined for every natural number.

Remark 2.2. Since $m$ is arbitrary, $n+m$ is defined for all $n,m\in \mathbb{N}$ .

2.1 Commutativity

Claim: $n+m=m+n$ for all $n,m\in \mathbb{N}$ .

We prove this by induction on $n$ .

Lemma 2.1.1. $0+m=m+0$ . (Proof by induction on $m$ ; left as an exercise.)

Lemma 2.1.2. $(n+m)++=n+(m++)$ . (Proof by induction on $m$ ; left as an exercise.)

Assuming $n+m=m+n$ , we compute:

2.2 Associativity

Claim: $(a+b)+c=a+(b+c)$ for all $a,b,c\in \mathbb{N}$ .

Proof. By induction on $a$ . $◻$

3. Multiplication and Exponentiation

Multiplication and exponentiation are defined by the same recursive pattern.

3.1 Multiplication

Set $n\times 0:=0$ .
If $n\times m$ is defined, set $n\times (m++):=n\times m+n$ .

3.2 Exponentiation

Set $a^0:=1$ (including the convention $0^0:=1$ ).
If $a^n$ is defined, set $a^{n++}:=a^n\times a$ .

The proofs of commutativity, associativity, and distributivity follow by repeated induction and are readily found in the literature. The purpose of this note is not to catalogue those arguments, but to illustrate how much structure can be derived from so few assumptions — and how recursive axioms such as Axiom 1.5 unlock the door to increasingly sophisticated algebraic constructions.

It is worth noting, finally, that modern foundations do not actually start from the Peano axioms. Instead, one typically begins with ZF or ZFC (the latter appending the Axiom of Choice) and constructs the natural numbers as sets via the von Neumann ordinals.