# Show the original Ackermann function is non-primitive recursive

There are a couple of questions here which show that the modern Ackermann function $$A(i, x)$$ is not primitive recursive. This new Ackermann function defined by Péter is a simplification of the original function proposed by Ackermann. I want to show that this original function is not primitive recursive. Unfortunately, the original paper is only in German (from what I could find) so I set about to try it myself.

He defines the function but the following double recursion.

$$\begin{cases} \zeta(0,b,a) = a+b \\ \zeta(n', 0, a) = \alpha(n,a)\\ \zeta(n',b', a) = \zeta(n,\zeta(n',b,a),a) \end{cases}$$ where $$\hspace{-.75in}\alpha(n,a) = \begin{cases} 0 \,\,\,\,\text{if}\,\,n= 0\\ 1\,\,\,\,\text{if}\,\,n =1\\ a\,\,\,\,\text{otherwise} \end{cases}.$$

For example, $$\zeta(0,b,a)=a+b$$, $$\,\,\,\,\zeta(1,b,a)=ab$$, $$\,\,\,\,\zeta(2,b,a)=a^b$$, $$\,\,\,\,\zeta(3,b,a)=a^{a^{a^{a}}}$$, with $$b$$ exponents. This creates a sequence of functions, iterating over $$n$$, each that grow fast than the functions that come before.

Now the proofs for the modern Ackermann function go like this. Define a set $$D$$ of number theoretic functions that are dominated by the function $$A(i,x)$$. Usually it looks like $$D = \{\phi:\exists t \,\,\text{s.t.}\,\, \phi(x_1, ...,x_n) < A(t, \max(x_k))\}$$. Then show that the successor function, the constant functions and the identity functions are in $$D$$. Lastly show that $$D$$ is closed under both schemata of primitive recursion. Then $$D$$ contains every primitive recursive function and if $$A(i,x)$$ were primitive recursive, it would dominate itself which is absurd.

I'm mainly wondering how I would define $$D$$ for the function given above. I started by defining $$\zeta(a) = \zeta(a, a, a).$$ This function $$\zeta(a)$$ then gives me the $$a$$th iteration of the sequence of functions with $$a,a$$ as arguments. Then I tried the sets $$D=\{\phi: \exists t \,\,\text{s.t.}\,\,\forall x_k > t, \phi(x_1,...,x_n) < \zeta(\max(x_k))\} \tag{1}$$ and $$D=\{\phi: \exists t,\,\,\exists x_k > t\,\,\text{s.t.}\,\, \phi(x_1,...,x_n) < \zeta(\max(x_k))\}. \tag{2}$$

However, with $$(1)$$ I couldn't show that $$D$$ was closed under the composition of functions schema, and with $$(2)$$ I couldn't show that the identity function, $$\phi(x_1,...,x_n) = x_k$$ for some $$k=1, ...,n$$ was a necessarily in $$D$$. My question is, what kind of $$D$$ should I set up to show it is closed under primitive recursive schema?

$$A(m,n)=\zeta(m-2,2,n+3)-3$$