Axioms:
1) $0$ is a counting number;
2) If $y$ is a counting number, so is $Sy$.

Definitions:
3) $x$ is an additive number in case for all counting numbers $y$, $x + y$ is also a counting number;
4) $x$ is a multiplicative number in case for all additive numbers $y$, $x \cdot y$ is also an additive number;
5) $x$ is a exponentiable number in case for all multiplicative numbers $y$, $x^y$ is also a multiplicative number;

Theorems:
6) If $x$ and $z$ are additive numbers, so is $x + z$.
7) If $x$ and $z$ are multiplicative numbers, so is $x \cdot z$.

Is it the case, that
8) If $x$ and $z$ are exponentiable numbers, so is $x^z$.

NO!