How to understand Smullyan's notion of weakly adequate Gödel numbering

Question

I was working on Smullyan's Diagonalization and Self-Reference (1994). In Chapter 9 (p.168), he defines the notion of a weakly adequate Gödel numbering as follows:

Let $\mathscr{L}$ be a first-order language and $g$ be a Gödel numbering. $g$ is weakly adequate if the following two conditions hold.

1. There is a relation $C(x,y,z)$ expressible in $\mathscr{L}$ such that for any expressions $X$ and $Y$ with respective Gödel numbers $x$ and $y$ the relation $C(x,y,z)$ holds if and only if $z$ is the Gödel number of $XY$.

2. There is a relation $N(x,y)$ such that for any Gödel number $x$, the relation $N(x,y)$ holds if and only if $y=\bar{x}$.

The first condition seems okay. To put it roughly, it means that the syntactic relation "$Z$ is the concatenation of $X$ and $Y$" is expressible in $\mathscr{L}$, where $Z$ is an expression with the Gödel number $z$.

But I have a trouble understanding the second condition. Here is why. Smullyan uses $\bar{x}$ to refer to the numeral denoting $x$ in $\mathscr{L}$. But what does it mean for a number to be a symbol? So it seems to me that the condition 2 should be rephrased as

2'. There is a relation $N(x,y)$ such that for any Gödel number $x$, the relation $N(x,y)$ holds if and only if $y$ is the Gödel number of $\bar{x}$.

Am I missing something here? Thanks for your help in advance!

Answer 1

See page 76 :

For any natural number $n$, by $\overline n$ we mean the symbol $0$ followed by $n$ accents [i.e. the term referring to the $n$-th successor of $0$], and we refer to $\overline n$ as the numeral designating $n$. [e.g., $0''''$ (sometimes used : $SSSS0$ ) is the numeral designating the number $4$.]

Having said that, IMO you are right; compare page 85 :

there is a function $\text {num}(x)$ such that for every number $x$, the number $\text {num}(x)$ is the $g$-number of the numeral $\overline x$.

If so, I agree with your proposed correction :

... the relation $\text N(x,y)$ holds if and only if $y$ is the $g$-number of the numeral $\overline x$.

Relations like $C(x,y,z)$ are "numerical" objects: they hold between numbers.

The trick of "arithmetization" is to achieve self-reference showing how to express or represent numerical relations with formulas of the language.