Exercise of "Gödel's Incompleteness Theorems" by Raymond Smullyan

Question

Is there a correct solution to Exercise 1 of Chapter 1 of the book "Gödel's Incompleteness Theorems" by Raymond Smullyan? I think it's impossible to find one.

The exercise:

Suppose L is a correct system such that the following two conditions hold: 1. The set P* is expressible in L; 2. For any predicate H, there is a predicate H' such that for every n, the sentence H'(n) is provable em L if and only if H(n) is refutable in L. Prove that L is incomplete.

Observations: P is the set of Godel numbers of provable sentences. P* is the set of numbers whose diagonal belongs to P.

My attempt:

$H'(n) \in \textbf{P} \iff H(n) \in \textbf{R} , \forall n$

The numbers h and h' are Godel number of H e H'. So:

$H'(h') \in \textbf{P} \iff d(h') \in P \iff h' \in P^* $

$H(h) \in \textbf{R} \iff d(h) \in R \iff h \in R^* $

If P* is expressible in L by a predicate G:

G(n) is true $\iff$ n $\in$ P*

And I am stuck

Answer 1

Let $E_n$ express $P^*$. Then for all $k$, $E_n(k)$ is true if and only if $E_k(k)$ is provable. Now say $E'_n= E_m.$ Then letting $k=m,$ $E_{n}(m)$ is true if and only if $E_{m}(m)$ is provable. But since $E_m=E_n',$ $E_m(m)$ is provable if and only if $E_n(m)$ is refutable. Thus $E_n(m)$ is true if and only if it is refutable.

Answer 2

Let $H$ be a predicate that expresses $P^{\ast}=\left\{n:g\left(E_n(n)\right)\in P\right\}$, where $P$ denotes the Godel numbers of all provable sentences. By assumption $H'$ is a predicate such that $H'(n)$ is provable iff $H(n)$ is refutable, for all $n$. Now, $H\left(g(H')\right)$ is true iff $g(H')\in P^{\ast}$ iff $g\left(H'\left(g(H')\right)\right)\in P$ iff $H'\left(g(H')\right)$ is provable iff $H(g(H'))$ is refutable.

Thus, $H(g(H'))$ cannot be true since $\mathscr{L}$ is assumed correct, so $H(g(H'))$ is false (and thus not provable) and not refutable.