axiomatic sets 18 (arithmetic in natural numbers 2)

The multiplication of natural numbers is defined same as that of the addition.

The operation $g$ of the multiplication is a function satisfying following conditions.
(You must remember the function $g$ is also a set.)

(1)$g(a,0)=0$
(2)$g(a,b^+)=g(a,b)+a$
 ,where $a,b\in\mathbb{N}$  and $b^+$  is the successor set of $b$ . ($b^+=b\cup\left\{b\right\}$)

These mean simply ;

$a\times 0=0$ ,
$a\times (b+1)=(a\times b)+a$ .

For example,
$g(2,3)=2\times(2+1)=(2\times 2)+2=((2\times 1)+2)+2$
 $=(((2\times 0)+2)+2)+2=0+2+2+2=6$ .

In addition, the exponent of natural numbers is extended.
By using the multiplication, the function $e$ of the exponent satisfies following conditions.

(1)$e(a,0)=1$ ,
(2)$e(a,b^+)=g(e(a,b),a)$ .

That is,
$a^0=1$ ,
$a^{b+1}=(a^b)\times a$ .

If $a=2,b^+=3$ ,then
$e(2,3)=g(e(2,2),2)=g(g(e(2,1),2),2)=g(g(g(e(2,0),2),2),2)=1\times 2\times 2\times 2$ .

These are called a finite recursion formula.