Paeno's Axioms:

1. There exist a number 'zero' in the set of all Natural Numbers

**N**
2. If a number

*a* is in

**N**, then its successor, S(

*a*), is also in

**N**
3. There is no

*a* in

**N** whereby the function S(

*a*) is equal to zero

4. For all

*a* and

*b* in

**N**, the function S(

*a*)=/=S(

*b*) if

*a*=/=

*b*
5. Suppose that

**S** is a subset of

**N**, such that zero exist in

**S** and that for any arbitrary

*a* which exist in

**S** has a successor S(

*a*) which exists in

**S**, then

**S** is equivalent to

**N**
Note: From thess axioms, we can define all natural numbers, for example, 1 = S(0), 2= S(1)= S(S(0)), 3 = S(2)=S(S(S(0))), etc.

Definition:

Iteration- Let there be a function

*f* which takes any element

*a* in the set

**S** from set

**S** to set

**S**
*f*:

**S** -->

**S**
An iteration of this function would be the composition of this function onto itself, or

*f*(

*f*(x)) where x exists in

**S**
Then the nth iteration would be

*f*^n(x) where by the profess of composing that function onto itself would be repeated n times.

Axiom-

*f*^n:

**S** -->

**S** , and satisfies the following conditions:

*f*^0 is the identity function in the set S, and

*f*^(S(n))(x)=

*f*(

*f*^n(x))

Transitive Property:

if a=b, and b=c, a must equal to c

Definition of Addition:

Let

*m,n* exist in set

**N**,

let the function S be the successor function

Let the function S^n represent the successor function S iterated n times.

m+n=S^n(m)

Definitions:

Numbers are defined as they are commonly used, the basis of those definition which shall be the above.

Prove: 5 + 7 = 12

1. Definition of Addition

5 + 7= S^7(5)

2. S^7(5)=S(S(S(S(S(S(S(S(5)))))))

3. S(5)=6 by definition of 6, so, S(S(S(S(S(S(S(S(5)))))))=S(S(S(S(S(S(S(6)))))))

4. S(6)=7 by virtue of the definition of 7, so S(S(S(S(S(S(7))))))=S(S(S(S(S(S(S(6)))))))

This process can be repeated until we at last come to

S(11)=12, therefore 12, by virtue of transitivity, must equal S^7(5), in other words, be equal to 5+7, so 5+7=12.

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