understanding gödel's 1931 paper - primitive recursive functions "FR(x)" and "n Gl x"

Question

I am trying very hard to fully understand gödel's paper on the incompleteness theorem.

I have a slight technical question about one of the 45 primitive recursive functions introduced to build the predicate (or "function" ? "relation" ?) provable(x).

http://mrob.com/pub/math/goedel.html

my question is : in the 22nd function

FR(x) ≡ (n){0 < n ≤ l(x) → Elf(n Gl x) ∨ (∃p,q)[0 < p,q < n & Op(n Gl x,p Gl x,q Gl x)]} & l(x) > 0

apparently in that formula we test for all n if the n-th item is an elementary formula or one obtained using the operators '¬', '∨' or '∀' on p,q-th formulae with p,q < n.

My question is : n Gl x ≡ ε y [y ≤ x & x/(n Pr x)y & not x/(n Pr x)y+1] where "n Pr x" is the n-th prime factor of x. And for a prime factor (or number) is associated the gödel number of a symbol only.

from what i understand about gödel numbering, "n Gl x" is only supposed to give a symbol form the alphabat the system is built on, not a formula.

What did i not understand ? How can "Elf(n Gl x)" or "Op(n Gl x,p Gl x,q Gl x)" possibly be true since, once again, from my understanding, "n Gl x" is only supposed to give the number of a symbol

Answer 1

Yes, $Bew(x)$ is a predicate; it holds if $x$ is the Gödel number of a provable formula $\varphi$.

In the 22nd formula, the predicate $FR(x)$ "tests" if the number $x$ is the code of a sequence of fomulae (and not only a sequence of "basic" symbols); it "return" $TRUE$ when for all $n : 0 < n \le lenght(x)$ the $n$-th item in the sequence is an elementary formula or it is obtained using the operators $¬, ∨$ or $∀$ from previous elements in the sequence, i.e. when there are $p,q < n$ such that ... .

See previous part of the document; the number of the "basic" symbols are used to assign number to strings of them (i.e. formula) and then to sequences of formulae. Thus, there is not only a uniquely associated natural number for every basic sign, but also for every sequence of basic signs.

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The formula 6 :

$n Gl x ≡ ε y [y ≤ x \land x/(n Pr x)^y \land \lnot x/(n Pr x)^{y+1}]$

can be used to "decode" a sequence.

Starting from a number $x$ encoding a sequence, it retrieves the greatest $y$ such that the $n$th prime number raised to $y$ divides $x$.

We can try with some simple examples; unfortunaltely, the original language used by Gödel does not use "$=$" as primitive.

We will assume it as primitive, and modify a little bit the basic encoding :

$0$ is "$=$"; $1$ is "$0$"; $3$ is "$f$"; $5$ is "$\lnot$"; $7$ is "$\lor$"; $9$ is "$\forall$"; $11$ is "("; $13$ is ")"; $17$ is "$x_1$", and so on...

In this way, we can encode the simple formula $0=0$ as : $2^13^05^1=2 \times 1 \times 5=10$.

Thus : $1Gl10=1, 2Gl10=0$ and $3Gl10=1$. Compulsing the alphabet, we can rewrite the string, because $1$ encodes "$0$" and $0$ encodes "$=$".

The encoding mechanism works due to the unique factorization into prime fcators; for every $n$ we can always retrieve with a finite number of divisions the exponents of its prime factors.

The encoding mechanism is used to encode not noly strings of symbols, but also sequences of strings : if $n$ is the code of the string $\sigma$, we encode with $2^n$ the sequence formed by the single string $\sigma$.

With the previous example, $2^{10}$ is the code of the sequence $\langle 0=0 \rangle$.

Now we can go on, considering a very simple example of proof in the system used by Gödel :

1) $\lnot(fx_1 = 0)$ --- Ax.1

2) $x_1 \forall \lnot (fx_1 = 0)$ --- from 1) by generalisation.

Let $n_1=2^53^{11}5^37^{17}11^013^117^{13}$ the code of the 1st formula, and let $n_2$ the code of the 2nd formula.

Thus, the number $n=2^{n_1}3^{n_2}$ is the code of the sequence :

$\langle \lnot(fx_1 = 0), x_1 \forall \lnot (fx_1 = 0) \rangle$.

The number $n$ encodes a proof in the system.