Revised edition
I'll use the translation in : Jean van Heijenoort, From Frege to Gödel : A Source Book in Mathematical Logic (1967), page 595-on.
[page 598] A formula of [the system] PM with exactly one free variable, that variable being of the type of the natural numbers (class of classes), will be called a class sign. [...] Let $\alpha$ be any class sign; by $[\alpha; n]$ we denote the formula that results from the class sign $\alpha$ when the free variable is replaced by the sign denoting the natural number $n$ [i.e. the numeral $S(S(\ldots S(0)))$].
[page 600] By $Subst \ a^v_b$ (where $a$ stands for a formula, $v$ for a variable, and $b$ for a sign of the same type as $v$ [a term]) we understand the formula that results from $a$ if in $a$ we replace $v$, wherever it is free, by $b$.
[page 603] The functions $x + y, x.y$, and $x^y$, as well as the relations $x < y$ and $x = y$, are recursive, as we can readily see. Starting from these notions, we now define a number of functions (relations) 1-45, each of which is defined in terms of preceding ones [...]
- $x/y \equiv (\exists z)[z < x \land x = y.z]$, $x$ is divisible by $y$.
This function is $/ : \mathbb N \to \mathbb N$, while e.g. $+ : \mathbb N \times \mathbb N \to \mathbb N$.
I.e. they are functions with natural numbers as "inputs" and "output".
[page 604] 13. $Neg(x) \equiv R(5)*E(x)$, $Neg(x)$ is the NEGATION of $x$.
Now we are in the "arithmetized" world of the syntax, but the functions are still numerical ones; thus, we have to read $Neg : \mathbb N \to \mathbb N$ as the function that takes as input the code of a formula $\varphi$ and produces as output the code of the formula $\lnot \varphi$.
I'll stay with the "standard" use of writing $\ulcorner \varphi \urcorner$ for the code of the formula $\varphi$.
- $Z(n) \equiv n N [R(1)]$, $Z(n)$ is the NUMERAL denoting the number $n$.
Again, a function that for each number $n$ calculates the code of the numeral $S(S(\ldots S(0)))$, due to the fact that $R(1)=2^1$ is the code of the sequence formed by the single symbol "$0$".
[page 605] $Su \ x^n_y$ results from $x$ when we substitute $y$ for the $n$th term of $x$.
[...] $Sb \ x^v_y$ is the notion $Subst \ a^v_y$ defined above.
$Subst$ is a syntactical operation performed on the formula $\varphi(y)$. The function $Sb$ is an arithmetical one, acting on number. Thus $Sb \ \ulcorner \varphi \urcorner ^v_y$ is the code of the formula $\varphi (v/y)$ obtained performing the $Subst$ operation.
[page 606 ] The fact that can be formulated vaguely by saying: every recursive relation is definable in the system $P$ (if the usual meaning is given to the formulas of this system), is expressed in precise language, without reference to any interpretation of the formulas of $P$, by the following theorem:
Theorem V. For every recursive relation $R(x_1, \ldots, x_n)$ there exists an $n$-place RELATION SIGN $r$ (with the FREE VARIABLES $u_1, u_2,\ldots, u_n)$ such that [...].
Thus, $R$ is a $n$-ary relation between numbers and $r$ is a formula of the formal system $P$.
[page 608] We now define the relation
$$Q(x,y) \equiv \lnot x B_k [Sb \ y^{19}_{Z(y)}].$$
$Q(x,y)$ is a binary numerical relation, defined in terms of previously inrtoduced functions and relations.
By Th.V, it is "expressible" in the system $P$ by a binary relation sign $q$, i.e. by a formula $q(x,y)$.
Let $\ulcorner q \urcorner$ be its code; it is the code number of a formula with two free variables, the two first var in the list of the alphabet, coded with $17$ and $19$ respectively.
Then, $\ulcorner p \urcorner = Gen(17,\ulcorner q \urcorner)$ is the code of the "universal closure" $\forall x q(x,y)$ of the formula $q(x,y)$ with respect to $x$, coded with $17$, due to the fact that the function $Gen(x,y)$ [page 604, n°14] calculates the code number of the GENERALIZATION of the formula coded by $y$ with respect to the VARIABLE coded by $x$.
Thus, the formula $p(y)$ has only one free variable.
Finally, we have $\ulcorner r \urcorner = Sb \ \ulcorner q \urcorner^{19}_{Z(\ulcorner p \urcorner)}$ where $r(x)$ is a new formula with one free variable obtained from $q(x,y)$ substituting the numeral $Z(\ulcorner p \urcorner)$ corresponding to the code for the formula $p(y)$ in place of the second free variable of $q(x,y)$.
This process is called diagonalization :
This is the idea of taking a wff $\varphi(y)$, and substituting (the numeral for) its own code number in place of the free variable. Think of a code number as a way of referring to a wff. Then the operation of ‘diagonalization’ allows us to form a wff that as it were indirectly refers to itself (refers to itself via the Gödel coding). We will use this trick [...] to form a Gödel sentence that encodes ‘I am unprovable in $\mathsf{PA}$’.
Following your translation, we have a formula $Q(x, y)$ that "means" : "$x$ does not prove the formula obtained from the formula with code $y$ by subst of the numeral $number(y)$ in place of the (only) free var".
Thus, $p = forall(17, q)$ means : "for all $x$, $x$ does not prove the formula obtained from the formula with code $y$ by subst of the numeral $number(y)$ in place of its only free variable", i.e. "the formula $\ldots$ is unprovable".
A very good book is :
See page 137 for some illuminating details :
[The expression] ‘$\ulcorner \varphi \urcorner$’ is shorthand for [the] standard numeral for the g.n. [Gödel number] of $\varphi$.
In other words, inside formal expressions ‘$\ulcorner \varphi \urcorner$’ stands in for the numeral for the number $\ulcorner \varphi \urcorner$.
A simple example to illustrate:
‘$SS0$’ is an expression, the standard numeral for $2$.
On our numbering scheme $\ulcorner SS0 \urcorner$, the g.n. of ‘$SS0$’, is $2^{21}.3^{21}.5^{19}$.
So, by our further convention, we can also use the expression ‘$\ulcorner SS0 \urcorner$’ inside (a definitional extension of) [the language], as an abbreviation for the standard numeral for that g.n., i.e. as an abbreviation for ‘$SSS \ldots S0$’ with $2^{21}.3^{21}.5^{19}$ occurrences of ‘$S$’ !
See also, for useful details : Diagonalization Lemma.
¬proofFor κ (x, subst(y, 19, number(y))) ⇒ provable κ (subst(q, 17 19, number(x) number(y)))
for some reason what i had in mind was : ¬proofFor κ (x, subst(y, 19, number(y))) ⇒ provable κ (subst(q, 17 19, x y))
Now it's all good ... and now i have to prove the theorem V ... i will probably post another question about that very subject
– joseph M'Bimbi-Bene Oct 25 '15 at 08:29