Tao's Real Analysis - Peano Axiom

We are forever friends.
Yes?
– Rhypho L.

0. Preface

Terence Tao presented in his Real Analysis script a very classical example of axiomatic system: Peano axiom, which formally answered some basic questions on natural numbers, like why 1+1=2.

Natural numbers are something that we know from the very beginning of education, but never ever strictly defined or thought about. This example gave the learner some insights on how to implement a mathematical system from a minimal dependency on axioms (things that we are believed in at the very beginning without any special reasons).

1. Peano axiom

Axiom 1.1

is a natural number.

forms the very basis of natural number system. Quite naturally should we count from , instead of (or other numbers we know), if we consider to be a natural number.

Also be careful that, we have not yet defined formally. And we should know that our theory is not limited to any specific symbol system, namely decimal, binary, octal etc.

We use to denote the operation to obtain the successor of a natural number. We want to say: form the natural number system. Herefore we introduce the following axiom.

Axiom 1.2

is a natural number, if is a natural number.

This axiom recursively generates the full natural number system.

Axiom 1.3

is not the successor of any natural numbers.

We would not count like: , as the computers do for the unsigned short. We believe that the natural number system is not cyclic.

Axiom 1.4

and do not yield the same object, if they are not the same.
In other words, two different objects cannot have the same successors.

Axiom 1.5: Principle of mathematical induction

Let be any property pretaining to a natural number . Suppose that is true, and suppose that whenever is true, is also true. Then is true for every natual number n.

The axiom 1.5 is a very powerful tool later on. It allows us to recursively define operations. Meanwhile, it makes our statement still true for countable infinity.

2. Addition

In this section, we introduce an operation called addition, denoted by . It is a binary operation which involves two natural numbers. To define recursively this new operation, we need the help of axiom 1.5. Recall it looks like:

1. Some property is true for ;
2. Assume that is true for ;
3. Consider . Is it well-defined? Does it give a true statement?

Definition 2.1

Assume to be a natural number, and we want to add it to another natural number.
For this, we define first .
We assume is defined.
And we define .

Examples 2.1.1:

Comment 2.1: We can view as a unary operation that increment exactly times now. By the induction step, it is defined for all the natural numbers.

Comment 2.2: Note that can be any natural number. Hence, is defined actually for any natural number and .

2.1 Commutativity

Does it hold that:

?

Yes! Use induction steps. Without loss of generality, we only do it for .

Lemma 2.1.1: clearly holds by doing another induction on . (exercise for reader).

Lemma 2.1.2 by doing another induction on . (another exercise for reader)

Suppose that , we consider :

2.2 Associativity

.

Proof: Induction on .

3. Multiplication and Power

Similarly, we can recursively define multiplication and power.

3.1 Multiplication

Define: .
If is already defined,
then define .

3.2 Power

Define: . Specially, define .
If is already defined,
then define .

In this article, I will not show you how to prove the properties like associativity, commutativity, distributivity etc. As you can easily find examples on the internet. This article is only meant to tell the readers how powerful an axiomatic system can be, even if it looks stupid and too simple at the beginning, and how to exploit the highly recursive axioms (like axiom 1.5) to discover the hidden facts and define more complicated algebras.

Also notably, the construction of modern analysis doesn’t really need Peano’s axiom. We start from ZFC (‘C’ stands for AC, axiom of choice), ZF, etc., and construct natural numbers from sets (von Neumann construction of number system).