TITLE: Concrete example for diagonal lemma
QUESTION [7 upvotes]: Diagonal lemma says that in a theory with enough assumption for any formula $A(x)$ there exist a sentence $B$ such that $B$ $\iff$ $A(\#(B))$ is a theorem in that theory, in which $\#(B)$ represents  the numeral of Gödel number of $B$.
I have no problem with the proof of this lemma, but I ask you to give me a concrete example for application of this lemma for a given formula $A(x)$ such as : $x$ is an even number.
I want this example to enhance my intuition for this lemma.
I want to ask you what is the intuitive idea behind this lemma? i.e. in spite of its mathematical proof, is this lemma expectable and intuitive for you that for any coding of formulas $B$ $\iff$ $A(\#(B))$ is a theorem in that theory or this lemma isn't expectable and intuitive? Please explain it more intuitively.
Thank you very much.

REPLY [3 votes]: I do not believe there is any easy intuition that would guide you to a proof of the diagonal lemma (e.g. the on on Wikipedia).  This proof is simply one of those things that seems to appear out of the blue; you find it amazing that the construction works; and after you learn the method you genuinely know more than you did before. 
Although the rudiments of the diagonal lemma are in Gödel's work, the diagonal lemma itself was first proved by Carnap, not by Gödel. More of the historical story, and references, are given in a comment by Peter Milne here. I think this emphasizes that even when the incompleteness theorems were first proven the diagonal lemma was not an obvious intuition. The lemma was only discovered later upon reflection. So there is no reason to think, as a student, that the lemma will be entirely intuitive at first.