Building product in $\Bbb N$ using the function $s: n\mapsto n+1$

Question

using the Peano's axioms we can give a description of the set of natural numbers.

Let's consider the functions

$s: \Bbb N\to \Bbb N$ defined by $s(n)=n+1$

and

$f^n=\begin{cases} id, &\text{ if } n=0 \\ f\circ f^{n-1},&\text { else} \end{cases}$

We can define the sum of two natural numbers in the following way: $n+m= s^n(m)$

How can I define the product using the power function and the function $s$?

Answer 1

Addition is defined as repeated application of the successor function $s$.

Similiarly, multiplication is defined as repeated addition.

If $n+m=s^n(m)\equiv \text{add}_n(m) = \text{add}_n \circ \text{add}_m(0)$, then we define multiplication by

$$n\cdot m = \text{add}_n^{m}(0) = (s^n)^m(0)$$

Answer 2

I tried defining operation in different ways.

Sum: $D(0)=m$ and $D(n+1)=D(n)+1$

$m+n= s^n(m)$

Product: $D(0)=m×0=0$ and $D(1)=m×1=m$ and $D(n+1)= D(n)+n$

$m×n=(s^n)^m= id$ if n=0 or m=0 else $\sum_{i=1}^{m}s^{n}$

Now that i defined the product i can say : $(s^n)^m$=s^(nm)

Power: $D(0)= m^0=1$ and $D(n)=D(n-1)×n$

$m^n= 1$ if n=0 else

$\sum_{i_1=1}^{m}(\sum_{i_2=1}^{m}(...(\sum_{i_n=1}^{m}s(0))))$ nested sum repeated n times

Are they correct? Would you write them differentely?