(I have only read second-hand about Gödel's incompleteness theorems, so I hope that there is nothing I have missed by doing that which would render my questions easily clarified.)
Gödel's first incompleteness theorem says that any consistent mathematical system is incomplete.
But the whole proof seems to depend solely on using a self-referential statement $G$ where
$G = $"G cannot be proven".
(I hope it is correct/ okay to put it like that)
It seems intuitive, however, that this statement is nonsensical.
$1.$ But then why is he allowed to use a nonsensical statement for proof?
To maybe explain at least a bit why I think it is nonsensical (if someone should not have that intuition strongly enough):
A. If we look at it as a definition of $G$, then it makes no sense to use $G$ itself in the definition of it.
This intuition could be partly the same as B.ii., but also $G$ doesn't add anything new to $G$. The self-referential aspect seems to make $G$ non-substantial.
So, $G$ seems to be without "real content".
B. Let $G_1= "G_2$ cannot be proven".
i. If $G_1$ and$ G_2 $are different, taken to be the same, this makes no sense (reminds of equivocation).
ii. If $G_1=G_2(=G)$, then we get an infinite regress.
Substitute $G_1 $for $G_2$, since it is the same thing, and we get ""$G_2$ cannot be proven" cannot be proven""= $G_1$. And do it again and we get " $G_2$ cannot be proven" cannot be proven" cannot be proven" =$ G_1.$ And so on infinitely.
In other words: "$G$ cannot be proven" can be infinitely substituted for $ G$ in "$G$ cannot be proven".
And maybe infinite regress (as a consequence) alone isn't enough to say of something that doesn't make any sense or should be discarded (we have Münchhausen Trilemma), but since $G$ intuitively also seems nonsensical (see A.), may it not be enough to discard it (/self-referential statements)?
- Also, if we do not use $G$ (or self-referential statements in general?), we would have no proof for any of what Gödel's incompleteness theorems show, is this correct? Or: if we discard $G$, then there is still the possibility that we can have consistent and complete and decidable systems?
Someone has said that there are other proofs for it using diagonalization. But if this is true, (since I also have intuitive issues with diagonalization arguments, though somewhat less), apart from these, there are none?
No. A standard Gödel sentence $G$ (for a given theory $T$) is NOT identical to the sentence "$G$ cannot be proven) (in $T$)".
But $G$ is so constructed that it is true if and only if $G$ cannot be proven in $T$.
Being materially equivalent and being identical are not the same. The difference is crucial here.
You don't need to read Gödel himself to understand that. There are many good expositions out there. For example, you could try the first three short chapters of my Gödel Without (Too Many) Tears, downloadable from https://www.logicmatters.net/igt