Reduction of Sums of Numerals to Numerals in E. Nelson's sense

Question

I have few questions about considerations from Edward Nelson's survey Warning Signs of a Possible Collapse of Contemporary Mathematics adressing dealing with countability & finiteness properties of numbers from ultrafinitistic point of view.

More precisely my concern adresses few details from following Sokratian style teacher student dialogue on page 8. It starts with:

S: Will you teach me Arithmetic?
T: Gladly, my boy. $0$ is a numeral; if $x$ is a numeral, so is $Sx$ (where $S$ is the successor function definied on class of all numerals (...not sure if Nelson regards numerals as a set at all).
Numerals are used to count things.
S: I understand.
T: Addition is introduced as follows:
$x + 0 = x$,
$x + Sy = S(x + y)$.
Try some addition.
S: I have encountered no difficulty in reducing sums of numerals to numerals.
T: Multiplication is introduced as follows:
$x \cdot 0 = 0$,
$x \cdot Sy = x + (x \cdot y).$
S: I've tried a few examples and reduced the products to numerals. But what about
$SS0 \cdot ... \cdot SS0 $
with $y$ occurrences of $SS0$, where $y$ is a long numeral?
T: It reduces to the numeral denoted by $SS0 \uparrow y$.
[...]

I'm struggling with precise meaning of both marked in bold statements.

Key Question: What does in this context precisely mean to "reduce sums of numerals to numerals"? What are the performed reduction steps in detail?

I know that Nelson's ultrafinitistic approach (compare with this discussion; especially in comments below Chow's & Nelson's answers) refuses any kind of inductive arguments/constructions and as known any naive attepts do define numerals iteratively like $n+1:= S(n)= S^{\circ(n+1)}(0)$ appears in Nelson's framework problematic as this obviously would face two potential problems:

(1): Either circularity: Namely, to say that $x$ is numeral iff $x=S^{\circ x}(0)$ is so far circular as $x$ is defined in terms including $x$ itself. This appears strange.

(2): Or undesired implicit reference to "classical" set of natural numbers $\Bbb N$ which as I guess Nelson would like to strictly avoid

So say $x,y$ are two numerals by definition above. How their sum $x+y$ in above sense reduces actually to numerals? Or, more specifically, which terms such expression is allowed to include except $0$ and $S$, and which not?

For instance, expressing $x+y$ as "numeral" $S^{\circ(x+y)}(0)$ would appear to be circular as remarked in point (1) and I guess this was not what Nelson would allow/wanted to see there. More generally, wouldn't any attept to reduce sum of numerals to a numeral run in similar circularity issues when trying to define recirsively the set of natural numbers refering finite quantities?

So which expression reducing $x+y$ to numeral was actually asked for? $S^{\circ y}(x)$? If yes, isn't there again appearing implicitly similar kinds of circularity in definition as indicated before in (1)? Moreover, why $x$ as numeral in Nelson's sense should be actually always given of shape $x=S^{\circ x}(0)$? Can it be concluded from definition without any induction principle? I don't see how.

But then - if from above reasons $S^{\circ y}(x)$ was not valid expression reducing sum of numerals $x,y$ to a numeral - after which form was else asked for?

Similarily, what is actually meant by reduction of products to numerals without running into circularity issues as before?

NB: At all my question appears to be quite pedantic, but exactly here - what actually the reduction of sums of numerals to numerals in Nelson's sense should be - I'm failing to see to consistency behind the approach Nelson proposes

Answer 1

Is $S(S(0))$ a numeral? Let's check.

$0$ is a numeral. Since $0$ is a numeral, $S(0)$ is a numeral. Since $S(0)$ is a numeral, $S(S(0))$ is a numeral. Great!

Is $S(S(0))+S(0)$ a numeral? Let's check.

Everything that's a numeral has the form $0$ or the form $S(x)$ where $x$ is a numeral. Since the expression $S(S(0))+S(0)$ has neither of these forms, it is not a numeral.

But $S(S(0))+S(0)$ is an expression involving $+$, and we have a definition of $+$. Let's apply it.

$S(S(0))+S(0)$ has the form $x+Sy$, where $x$ is $S(S(0))$ and $y$ is $0$. So $S(S(0))+S(0)$ reduces to $S(S(S(0))+0)$.

Now the inner expression $S(S(0))+0$ has the form $x+0$, where $x$ is $S(S(0))$. So $S(S(S(0))+0)$ reduces to $S(S(S(0)))$, which is a numeral.

More compactly, we could write the reduction steps as $$S(S(0))+S(0)\rightarrow S(S(S(0))+0) \rightarrow S(S(S(0)))$$

Note that we have not introduced any general rule here like $S^{x}(0)+S^y(0) = S^{x+y}(0)$. We're simply taking an expression involving $+$, pattern matching, and applying the definition until we arrive at a numeral.

When S says "I have encountered no difficulty in reducing sums of numerals to numerals", he means just this: in all the examples S has tried, expressions of the form $x+y$, where $x$ and $y$ are numerals, have indeed reduced to numerals by following the pattern matching algorithm described above.

Later, when T talks about "the numeral denoted by $S(S(0))\uparrow y$", he's not saying that $S(S(0))\uparrow y$ is a numeral - it's not! He's saying that there is a particular numeral which is named by the expression $S(S(0))\uparrow y$ in the same way that the expression $S(S(0))+S(0)$ names the numeral $S(S(S(0)))$.