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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?

spacemonkey
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  • Not very clear... but 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 $s(s(\ldots (0) \ldots))$ i.e. strings of $n$ times the sucecssor function applied to the term $0$. – Mauro ALLEGRANZA Mar 08 '24 at 10:15
  • Thus, when we substitute a code number $u$ in a formula, what we "really" perform is the substitution of the symbol $x$ (a variable, i.e. a term) with a new symbol $s(s(... (0)...))$ that is again a term: the numeral "naming" the number $u$. – Mauro ALLEGRANZA Mar 08 '24 at 10:17
  • With "we want to decode this number back into some formula" what do you mean? 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. – Mauro ALLEGRANZA Mar 08 '24 at 10:20
  • See example and see (maybe) relevant post. – Mauro ALLEGRANZA Mar 08 '24 at 10:21
  • See also this post for more details. – Mauro ALLEGRANZA Mar 08 '24 at 10:23
  • 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 very big, 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 $(sssss0=0)$ that you can encode again. – Mauro ALLEGRANZA Mar 08 '24 at 10:27
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    @MauroALLEGRANZA sorry for the late reply. You are completely right, I forgot that we are dealing with numerals. It makes sense now, thanks! – spacemonkey Mar 10 '24 at 19:58

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