In Mendelson's "Number Systems" he defines and subsequently proves the existence and uniqueness of binary operators $+$ and $\times$, as well as exponentiation, in $P$ using the so-called Iteration Theorem:
Consider any Peano system ($P$, $S$, $1$). Let $W$ be an arbitrary set, let $c$ be a fixed element of $W$, and let $g$ be a singulary operation on $W$ (that is, $g:W \to W$). Then, there is a unique function $F: P \to W$ such that
- $F(1) = c$.
- $F(S(x)) = g(F(x))$ for all $x$ in $P$.
He defines addition and multiplication as follows:
Consider any Peano system ($P$, $S$, $1$). Then there is a unique binary operation $+$ on $P$ such that
- $x + 1 = S(x)$ for all $x$ in $P$.
- $x + S(y) = S(x + y)$ for all $x$ and $y$ in $P$.
Consider any Peano system ($P$, $S$, $1$). Then there is a unique binary operation $\times$ on $P$ such that
- $x \times 1 = x$ for all $x$ in $P$.
- $x \times S(y) = (x \times y) + x$ for all $x$ and $y$ in $P$.
I denote addition, multiplication, and exponentiation respectively by $a_1, a_2, a_3$, and in general define $a_n$ for $n > 1$ as
For any Peano system ($P$, $S$, $1$), there exists a unique binary operation $a_n$ such that
- $a_n(x, 1) = x$ for all $x$ in $P$.
- $a_n(x, S(y)) = a_{n-1}(x, a_n(x,y))$ for all $x,y$ in $P$.
How can we prove the existence of the operation $a_n$ for all natural $n > 1$? In addition, how can we prove the non-commutativity of $a_n$ if $n > 2$, e.g., $a_4(4, 5) = {^{5}4} \neq a_4(5,4) = {^{4}5}$?
