It is well known that [Ackermann's function](https://en.wikipedia.org/wiki/Ackermann_function) is not primitive recursive. Therefore, the theories of primitive recursive arithmetic (PRA) and of $\Sigma\_1$-induction ($I\Sigma\_1$) cannot prove the totality of Ackermann's function.


However $\Sigma\_2$-induction suffices to prove the totality of Ackermann's function. It also proves the consistency of $I\Sigma\_1$.


My question: Is already the totality of Ackermann's function strong enough to prove the consistency of PRA or $I\Sigma\_1$?



Does the totality of Ackermann's function prove the consistency of $\Sigma_1$-induction?
The answer is apparently yes.


Recall the Grzegorczk hierarchy $\mathcal{E}^n$. This a hierarchy which classifies computable functions based on their growth rate, see [Wikipedia: Grzegorczyk hierarchy](https://en.wikipedia.org/wiki/Grzegorczyk_hierarchy).


Now on this hierarchy can be extended to ordinal level, see [Wikipedia: Fast Growing Hierarchy](https://en.wikipedia.org/wiki/Fast-growing_hierarchy).
Then we have $\mathcal{E}^\omega:= \bigcup\_{n<\omega} \mathcal{E}^n$, which is the same as the set of all primitive recursive functions.
On the next level $\mathcal{E}^{\omega+1}$ one find then a variant of the Ackermann function.


Now Cleave, Rose formulate in [$\mathcal{E}^n$ arithmetic](http://zbmath.org/?q=an:03259892) an arithmetical systems $\mathcal{E}^n$ corresponding to the set of functions $\mathcal{E}^n$. Further, they show there that $\mathcal{E}^{n+1}$ proves the consistency of $\mathcal{E}^n$ for $n<\omega$.
In [$\mathcal{E}^\alpha$ and transfinite induction](http://zbmath.org/?q=an:0262.02043) Rose claims that this result is also true for $n$ being an infinite ordinal.
Thus arithmetic with a variant of the Ackermann function proves consistency of PRA (which is the same as consistency of $I\Sigma\_1$).


