TITLE: Gödel's ontological proof
QUESTION [42 upvotes]: Gödel's ontological proof is a formal argument for God's existence by the mathematician Kurt Gödel.
Can someone please explain what are the symbols in the proof and elaborate about its flow:
$$
\begin{array}{rl} \text{Ax. 1.} & \left\{P(\varphi) \wedge \Box \; \forall x[\varphi(x) \to \psi(x)]\right\} \to P(\psi) \\ \text{Ax. 2.} & P(\neg \varphi) \leftrightarrow \neg P(\varphi) \\ \text{Th. 1.} & P(\varphi) \to \Diamond \; \exists x[\varphi(x)] \\ \text{Df. 1.} & G(x) \iff \forall \varphi [P(\varphi) \to \varphi(x)] \\ \text{Ax. 3.} & P(G) \\ \text{Th. 2.} & \Diamond \; \exists x \; G(x) \\ \text{Df. 2.} & \varphi \text{ ess } x \iff \varphi(x) \wedge \forall \psi \left\{\psi(x) \to \Box \; \forall y[\varphi(y) \to \psi(y)]\right\} \\ \text{Ax. 4.} & P(\varphi) \to \Box \; P(\varphi) \\ \text{Th. 3.} & G(x) \to G \text{ ess } x \\ \text{Df. 3.} & E(x) \iff \forall \varphi[\varphi \text{ ess } x \to \Box \; \exists y \; \varphi(y)] \\ \text{Ax. 5.} & P(E) \\ \text{Th. 4.} & \Box \; \exists x \; G(x) \end{array} $$
Does it prove both existence and uniqueness?
Edit: these are modal logic symbols.

REPLY [2 votes]: I think the most important matter in understanding Gödel's proof is the interpretation of $P$. In this respect, I would like to mention that the popular interpretation of $P(φ)$ as "$φ$ is positive" is not entirely accurate, as a result of axiom $1$. Taking both axioms $1$ and $2$ into account, it is best interpreted as "$φ$ is a property that an all-positive entity has". The negation of $P(φ)$ does not mean "$φ$ is negative" and neither does it mean "$φ$ is a property that an all-negative entity has". It means either "$φ$ is not a property that an all-positive entity has", or "$φ$ is a property that an all-positive entity does not have". (These are equivalent, which means the proposed interpretation conforms to axiom $2$. There cannot be more than one way to not have a property, as opposed to the possibilities neutral and negative for the "positive"-interpretation.)
Consider the property $ψ_1(x)$ to mean "$x$ is extreme in politeness", and consider that this means "either $x$ is very polite or $x$ is very impolite". This property is rather neutral than positive, yet, it is a property that an all-positive entity has. Indeed, $\neg ψ_1(x)$ would imply that $x$ is not very polite (apart from the fact that $x$ is not very impolite either). Note that $ψ_1$ is a property that also an all-negative entity has. We could call $ψ_1$ a two-edged property.
The given interpretation is indeed inspired by axiom $1$. Consider the property $φ_1(x)$ to mean "$x$ is very polite" and consider that $P(φ_1)$ holds. This $φ_1$ forces $ψ_1$ (it is necessarily true that anything with property $φ_1$ has property $ψ_1$). Now it follows from axiom $1$ that $P(ψ_1)$ holds, even though, as argued above, $ψ_1$ is not more positive than it is negative. Also an all-negative entity has it. A possible solution is to interprete $P(φ)$ as: "$φ$ is a property that is held by an all-positive entity". (I let it to the creativity of the reader to find an interpretation that conforms to axiom $1$, but not to axiom $2$.)
$G(x)$ means "$x$ is an all-positive entity", or "$x$ is God-like". Gödel proves that such an entity necessarily exists. The difference in interpretation of $P$ is of importance in analyzing whether the converse of the proof-scheme also proves that an all-negative entity necessarily exists. We could interprete $P(φ)$ to mean "$φ$ is a property that an all-negative entity has" and $G(x)$ to mean "$x$ is Devil-like". The truth of axiom $5$ ($P(E)$ holds) can especially depend on a correct interpretation of $P$. If $P(φ)$ meant just "positive" (conversely "negative"), then existence, according to axiom $2$, could not be a property that can be held by both a God-like and a Devil-like entity. Considering that $P(φ)$ means "$φ$ is a property that an all-positive (conversely all-negative) entity has", it is possible that existence is a two-edged property: both God-like and Devil-like entities can have it. Since a Devil-like entity that exists is more negative than a Devil-like entity that fails to exist, it appears that the converse of axiom $5$ also holds: existence is a property held by an all-negative entity. Therefore, according to Gödel's axioms, it is also necessary that a Devil-like entity exists.