Ok, so I'm reading the chapter on Gödel's Incompleteness Theorem in "Gödel, Escher, Bach" and I want to make sure I'm getting this right: the idea of the book's proof is to form the sentence
"There exists no $a,a'$ with both (1) $a$ being the Gödel number of the derivation of the sentence having $a'$ as its Gödel number, and (2) $a'$ being the Gödel number of the sentence
"There exists no $a,a'$ with both (1) $a$ being the Gödel number of the derivation of the sentence having $a'$ as its Gödel number, and (2) $a'$ being the Gödel number of the sentence produced by substituting the free variables of the sentence with Gödel number $u$ by $u$"",
where $u$ is the Gödel number of the sentence
"There exists no $a,a'$ with both (1) $a$ being the Gödel number of the derivation of the sentence having $a'$ as its Gödel number, and (2) $a'$ being the Gödel number of the sentence produced by substituting the free variables of the sentence with Gödel number $a''$ by $a''$",
$a''$ being free. Since this $a'$ actually exists, (we do indeed have a sentence to Gödel number), then the whole thing is the same as
"There exists no $a$ with $a$ being the Gödel number of the derivation of the sentence
"There exists no $a,a'$ with both (1) $a$ being the Gödel number of the derivation of the sentence having $a'$ as its Godel number, and (2) $a'$ being the Godel number of the sentence produced by substituting the free variables of the sentence with Gödel number $u$ by $u$""
But the sentence produced by substituting the free variables of the sentence with Gödel number $u$ by $u$ is
"There exists no $a,a'$ with both (1) $a$ being the Gödel number of the derivation of the sentence having $a'$ as its Gödel number, and (2) $a'$ being the Gödel number of the sentence produced by substituting the free variables of the sentence with Gödel number $u$ by $u$",
so that we get the original sentence is equivalent to
"There exists no $a$ with $a$ being the Gödel number of the derivation of the sentence
"There exists no $a,a'$ with both (1) $a$ being the Gödel number of the derivation of the sentence having $a'$ as its Gödel number, and (2) $a'$ being the Gödel number of the sentence
"There exists no $a,a'$ with both (1) $a$ being the Gödel number of the derivation of the sentence having $a'$ as its Gödel number, and (2) $a'$ being the Gödel number of the sentence produced by substituting the free variables of the sentence with Gödel number $u$ by $u$.""";
but the last two paragraphs are exactly the original sentence, so we finally get
"There exists no $a$ with $a$ being the Gödel number of the derivation of this sentence".
So this sentece must be true (otherwise we get a contradiction) and therefore not provable.
Is this all right?