Differences between between concepts related to Gödel's Incompleteness theorems: self-referencing, diagonalization and fixed point theorem?

Question

I am studying the proof of Gödel's first Incompleteness theorem at the moment and I don't understand the differences between self-referencing, diagonalization and fixed point related to Gödel's proof.

In my opinion, they all mean taking the Gödel number of a formula as an argument into the formula itself.

Can someone please explain me the differences?

Thanks in advance.

Answer 1

It would probably help if you provide instances of each. But if you're just talking about their use in Gödel incompleteness theorems, then I suppose they all mean different aspects of the same thing. Application of the fixed point lemma is often referred to as diagonalization (of a given formula), and then that formula is said to be referring to itself.

I recommend:

Raymond Smullyan, 1994. Diagonalization and Self-Reference. Oxford Univ. Press.

Raymond Smullyan, 1991. Gödel's Incompleteness Theorems. Oxford Univ. Press.