Decoding Gödel's number inside Gödel's number

Question

Since Gödel's numbering maps formulas into numbers, how would a number inside a number that represents a formula be decoded? Let's say I have the following formula of a formal system called TNT (from the book "Gödel, Escher, Bach" by Douglas Hofstadter):

¬∃a:∃b:<PROOF-PAIR(a,b)∧ARITHMOQUINE(c,b)>

It has some Gödel's number $u$, e.g.: 123,321,444,... If we replace the sequence of numbers in $u$ that represents a free variable $c$ with $u$ itself, we would get another Gödel's number -- G.

Now suppose we want to decode this number back into some formula of TNT. To what is the sequence in place of $c$ is decoded too? It can't be decoded into a number 123,321,444..., since numbers have their own codes. However, if it is decoded into our original formula, then we are passing a relation into a predicate. This means our decoded formula is not wff, no?

Answer 1

The best way to avoid conundrums with codes is to recall that when we speak in the metatheory of the code of a formula, this is a usual natural number $n$, while when we consider numerical formulas they are written using the formal terms of the language and we have no numbers but numerals, i.e. names for number that are formally expressions like: $s(s(…(0)…))$, i.e. strings of $n$ times the sucecssor function applied to the term $0$.

When we substitute a code number $n$ in a formula, what we really perform is the substitution of the term $x$ (a variable) occurring into the formula with a new symbol $s(s(...(0)...))$ that is itself a term: the numeral naming the number $n$.

What do you mean with "we want to decode this number back into some formula"?

In general, not every numebr is a code, but we have (i) a mechanical procedure to check if a number $n$ is a code, and (ii) if the answer is Yes, the same mechanical procedure gives us as output the formal expression: term, formula, etc encoded by $n$.

You can try some very simple exercise... but you have to use very short formulas, because numbers grow very fast: let $(x=0)$ the formula and compute its code $u$. Then you have to write the numeral corresponding to $u$; if $u$ is not huge, let it be $5$, the corresponding term will be (skipping parentheses) $sssss0$ and thus when you replace it in formula coded by $u$ what you get is simply the new formula $(sssss0=0)$.