# What is missing on Gödel's theorem explanation in “Emperor's New Mind” from Roger Penrose?

In "Emperor's New Mind", from pages 132 to 141, Roger Penrose explains Gödel's incompleteness theorem in very simple terms. Basically he says that, in a system of ordered propositions ($P_1(w)$, $P_2(w)$, ..., $P_n(w)$) and ordered proofs ($\Pi_1$, $\Pi_2$, ..., $\Pi_n$), the $k$th proposition is $P_k(w) = \nexists x [ \Pi_x \, proves \, P_w(w)]$. If we apply $k$ to $Pk(w)$ we will get $P_k(k) = \nexists x [ \Pi_x \, proves \, P_k(k) ]$, which is a Gödel-unprovable proposition.

That is a simple explanation, and it makes perfect sense to me, on the other hand Gödel original paper is very complicated, and there are a lot of discussion on the internet on how to understand and how to explain Gödel's theorem to laypeople, which means there is probably some more complicated part of the theorem that is absent of Penrose's explanation. Is that right? What is it?

(Penrose himself says there is a part of Gödel's argument that is super-complicated, but that it is only the part that explains and proves that it is possible to have mathematical proofs and propositions written using only numbers, no symbols -- with some numbers meaning symbols? -- and that these numbers can be operated and ordered like normal numbers, is that the answer to my question above? If yes, how important is this part to the total comprehension of the whole argument? It doesn't seem very important.)

As I am not experienced in these mathematical things, please forgive (and hopefully fix) my mistakes in this question.

1. The theory of the propositions ($$P_1(w)$$, $$P_2(w)$$, ..., $$P_n(w)$$) needs to be consistent. If this theory is not consistent then any proposition, including $$P_k(k)$$, is provable from the theory. As this would contradict Godel unprovability consistency needs to be (formalised and) assumed.
2. $$P_k(k)$$ might be unprovable because $$\lnot P_k(k)$$ is provable. Now $$\lnot P_k(k) = \exists x [ \Pi_x \, proves \, P_k(k) ]$$ stating that a proof of $$P_k(k)$$ exists, a proof whose index is $$x$$. Ruling this out required the assumption of $$\omega-$$consistency of the given theory (of arithmetic).