Definitions

1. Positive Property: A property P is said to be positive if and only if it satisfies certain abstract moral or aesthetic criteria. Let $Positive(P)$ denote this predicate.

2. God-like: $\varphi$ is God-like if and only if $\varphi$ possesses all positive properties:

3. Necessary Existence: The necessary existence of $\varphi$, denoted $NE(\varphi)$, is defined as:

Axioms

1. A property is either positive or its negation is positive:

2. A property necessarily entailed by a positive property is positive:

3. The property of being God-like is positive:

4. Positive properties are necessarily positive:

5. Necessary existence is a positive property:

Theorems and Proofs

Theorem 1: If God exists, then God exists necessarily.

From Axiom 3, $Positive(God)$. By the definition of God-like, a God-like entity possesses all positive properties. Since necessary existence $(NE)$ is a positive property (Axiom 5), a God-like entity necessarily exists.

Theorem 2: God exists necessarily in the S5 modal logic system.

In S5, if something is possibly necessary, it is necessary $(\Diamond \Box \varphi \to \Box \varphi)$. Assume $\Diamond \exists x (God(x))$. By the definition of God-like, $NE$ applies to a God-like entity, so $\Box \exists x (God(x))$. Therefore, $\exists x (God(x))$ holds necessarily.