Now let's build the natural integers line, the set N that is made of 0 plus the numbers we count with 1, 2, 3, etc.

The natural integers line starts with 0, that means no object, continues with the follower of 0, that is 1, and then iterates the follower determination to count.

The follower of 1 is 2, the follower of 2 is 3, etc.

The follower of 3 is 4, the follower of 4 is 5, 6, 7, 8, the follower of 8 is 9, the follower of 9 is 10, 11, 12, etc.

The follower of the k-th element is k plus 1.

Find the follower is adding 1.

The formal construction of the set N of natural integers is based on the Peano axioms.

First axiom: 0 is a natural integer.

Any natural integer has a unique follower that is a natural integer.

There is no end to this process.

Zero is not the follower of any natural integer.

It begins with zero.

Any non-zero natural integer is the follower of a unique natural integer.

There are no branches and it is a line.

The properties of the set N of natural integers is that it is built as a line starting from zero and with no end. It goes to infinity.

So we can say that we may count from zero to infinity.

What's coming now? What's coming now is an example of proof by recursion.

It is a powerful way to prove theorems based on natural integers construction.