Intuitively, we construct the natural numbers when we write down
expressions 0, 0+1, 0+1+1, and so on, and then
consider the set of objects that can be written down with this recipe.
This seems to contain a circularity, for the definition of our set of
natural numbers would seem to be be the collection of all
0+1+

Definition.

Choose __ to be some fixed infinite universe of objects, and
____
__

zeroor 0. Now let

where the big

For particularly suspicious readers, we should give an example of
a suitable __ and ____ to be the
collection of all sets, and __

Principle of Induction.

Suppose

is a property such that(0)

, i.e., 0 has property , and(

(

(

**Proof.**

Let __
that have property __

(__)__

(

Principle of Inductive Definitions.

A definition of the following kind,
_{0
}

**Proof.**

Let _{0}

There follows a long sequence of propositions proving (by induction) that
+ and

Proposition.

The following holds for all

**Proof.**

Induction on

Proposition.

The following holds for all

**Proof.**

Induction on

Proposition.

The following holds for all

**Proof.**

Induction on

Proposition.

The following holds for all

**Proof.**

Proposition.

The following holds for all

**Proof.**

Induction on

Proposition.

The following holds for all

**Proof.**

If __,
so 0 cannot equal __

Proposition.

The following holds for all

**Proof.**

Induction on

Proposition.

The following holds for all

**Proof.**

If __,
__ with

Proposition.

The following holds for all

**Proof.**

If __=
__

Proposition.

The following holds for all

**Proof.**

Induction on __
____
__. If

Theorem.

If

**Proof.**

Existence is by induction on __+1__
__
and ____+1)=
__

For uniqueness suppose,

Definition.

If

We also make the usual definitions concerning *least number principle*.

Least number principle.

Suppose
there is some

**Proof.**

Let

(

**Subproof.**

Suppose there is some

(

**Subproof.**

Suppose there is no least

(

Then 0

(

(

Also if

(

(

(

(

It follows by induction that

(

(

This proves by contradiction, that there is a least

(