I am trying to teach myself Gödel numbering from this SEP page: https://plato.stanford.edu/entries/goedel-incompleteness/sup1.html
I understand why the Gödel number of '0=0' is 2430 (in this coding system), but further on the article talks about negation: neg(⌈A⌉)=(⌈¬A⌉)
What is the Gödel number of ⌈¬A⌉? And why?
On
The general idea is simple: every "basic" symbol has a code (its own G-number).
A formula (i.e. an expression of the language) is a finite string of symbols.
Thus, a formula $abc$ (where $a,b,c$ are basic symbols) is encoded with $c = \ulcorner abc \urcorner = 2^{c_1} \times 3^{c_2} \times 5^{c_3}$, where $c_1 = \ulcorner a \urcorner$ is the code of the symbol $a$, and so on.
Thus, the code of $\lnot A$ is $2^9 \times 3^{\ulcorner A \urcorner}$.
For example, the G-number of $¬(0=0)$ will be : $2^9 \times 3^{2430}$.
First of all, where $A$ is a formula of logic (i.e a string of symbols), $\lceil A \rceil$ is the Gödel number of that formula. So notice that there is no such thing as 'the Gödel number of $\lceil \neg A \rceil$'; that's like asking what is the Gödel number of the Gödel number of $\neg A$. The Gödel numbering only applies to formulas (which may be the logical representation of numbers, but which aren't numbers themselves).
Second, the function $neg$ is simply defined as the function that maps the Godel number of any formula $A$ to the Godel number of the formula $\neg A$ ... it, by itself, does not tell you how to figure out what the Godel number of $\neg A$ is on the basis of the Godel number of $A$. For that, you simply go back to the idea of numbering strings/sequences of symbols as explained in the article. Thus, where $A$ is $0=0$, $\neg A$ is $\neg 0 = 0$, and so its Godel number is:
$$2^9 \cdot 3^1 \cdot 5^5 \cdot 7^1 = 33600000$$
Finally - and this has nothing to do with your question - the article seems to have a slight problem in that it defines $impl(\lceil A \rceil, \lceil B \rceil) = \lceil A \rightarrow B \rceil$ .... but that should probably be $impl(\lceil A \rceil, \lceil B \rceil) = \lceil ( A \rightarrow B ) \rceil$, for if you don't put parentheses around the conditional, then $neg(impl(\lceil A \rceil, \lceil B \rceil))$ would give the Godel number of $\neg A \rightarrow B$ instead of the desired $\neg (A \rightarrow B)$ ... you should send Panu an email about this!