(Not completed) Tao 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. Natural numbers is something that we know from the very beginning of education, but never ever strictly defined. This example gave the learner some insights on how to implement a mathematical system from a minimal dependency on axioms.

1. Peano axiom

Axiom 1.1

is a natural number.

forms the very basis of natural number system. Rather naturally should we count from , instead of , if we consider to be a natural number.

Also be careful that, we have not yet defined formally. And we would try the best to avoid using this symbol, because we want our theory to be as general as possible, i.e. not limited by any specific symbol system.

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.

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.

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 :

To be continued…