I have read Edward Nelson's *Warning signs of a possible collapse of contemporary mathematics* a couple of times, it is a very interesting read, but I do not understand the conclusory paragraph. In particular I am interested in the final 'warning sign', beginning from the fable on page 8 and continuing through to the end (all relevant information for my question is copied below).

The belief that exponentiation, superexponentiation, and so forth, applied to numerals yield numerals is just that—a belief.

Note: we are working in the standard 7 axioms of PA for this question.

*Numerals* are defined as follows. $0$ is a numeral; if $x$ is a numeral, so is $\text{S}x$. In a standard way, we introduce *addition*:
$$\begin{align}
x+0 &=x,\\
x+\text{S}y &=\text{S}(x+y),
\end{align}$$
*multiplication*:
$$\begin{align}
x\cdot 0 &=0,\\
x\cdot\text{S}y &=x+(x\cdot y),
\end{align}$$
and *exponentiation*:
$$\begin{align}
x\uparrow 0 &=\text{S}0,\\
x\uparrow\text{S}y &=x\cdot(x\uparrow y).
\end{align}$$
Then we define *counting numbers*, introducing two new axioms to our system:

- $0$ is a counting number.
- If $x$ is a counting number, then $\text Sx$ is a counting number.

*additionable numbers*:

$x$ is an additionable number in case for all counting numbers $y$, the sum $y + x$ is a counting number.

and finally, *multiplicable numbers*:

$x$ is a multiplicable number in case for all additionable numbers $y$, the product $y \cdot x$ is an additionable number.

We prove that additionable and multiplicable numbers are closed—i.e. if $x_1$ and $x_2$ are additionable (respectively, multiplicable) numbers, then $x_1+x_2$ is additionable (respectively, $x_1\cdot x_2$ is multiplicable).

But now we come to a halt. If we attempt to define 'exponentiable number' in the same spirit, we are unable to prove that if $x_1$ and $x_2$ are exponentiable numbers then so is $x_1 \uparrow x_2$. [...] The belief that exponentiation, superexponentiation, and so forth, applied to numerals yield numerals is just that—a belief.

It is at this point that I lose the author. If I understand correctly, he is saying that given two numerals $x_1$ and $x_2$ (i.e. $\text S\text S\text S\ldots 0=x_1$ for some number of $\text S$s, and similarly for $x_2$), then $$x_1\uparrow x_2=\underbrace{x_1\cdots x_1}_{x_2\text{-times}}$$ is not a numeral (i.e. we cannot definitely say that there is some number of $\text S$s for which $x_1\uparrow x_2=\text S\text S\text S\ldots 0$). If this is not what was intended, what is the author's point at the end here?

However, if I understand right, then I am not convinced; below I demonstrate my attempt to show this given some $x_1\uparrow x_2$. Can we not 'break this down' as follows? (From here I use the notation $\text S^k 0$ for the numeral $k$, that is the numeral $\underbrace{\text{SS}\ldots\text S}_{k\text{-times}}0$.)

$$\begin{align} x_1\uparrow x_2&=\underbrace{x_1\cdots x_1}_{x_2\text{-times}}\\ &=\underbrace{(\text S^{x_1} 0)\cdots (\text S^{x_1} 0)}_{(\text S^{x_2} 0)\text{-times}}\\ &=(\text S^{x_1} 0\cdot \text S^{x_1} 0)\cdot\underbrace{(\text S^{x_1} 0)\cdots (\text S^{x_1} 0)}_{(\text S^{x_2}0-2 )\text{-times}}, \end{align}$$ now apply the definition of multiplication to $\text S^{x_1} 0\cdot \text S^{x_1} 0$ to find $$\begin{align} \text S^{x_1} 0\cdot \text S^{x_1} 0&=\text S^{x_1} 0+(\text S^{x_1} 0\cdot \text S^{x_1-1} 0)\\ &=\text S^{x_1} 0+\text S^{x_1} 0+(\text S^{x_1} 0\cdot \text S^{x_1-2} 0)\\ &=\cdots=\underbrace{\text S^{x_1} 0+\text S^{x_1} 0+\cdots+\text S^{x_1} 0}_{x_1\text{-times}}+(\text S^{x_1} 0\cdot 0)\\ &=\underbrace{\text S^{x_1} 0+\text S^{x_1} 0+\cdots+\text S^{x_1} 0}_{x_1\text{-times}}. \end{align}$$ Do this a finite number of times to put $x_1\uparrow x_2$ in the form of an addition; at which point applying the definition of addition a finite number of times should yield $$x_1\uparrow x_2=\underbrace{\text{SS}\ldots\text S}_{x_1^{x_2}\text{-times}}0.$$ Now $x_1\uparrow x_2=\underbrace{\text{SS}\ldots\text S}_{x_1^{x_2}\text{-times}}0$ is a numeral by induction.

The claim 'that exponentiation, superexponentiation, and so forth, applied to numerals yield numerals is just that—a belief' is a strong one, what does Nelson mean and how does he conclude this?

Thank you!