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Godel's Ontological Proof Outline
The proof uses modal logic, which distinguishes between necessary truths and contingent truths. In the most common semantics for modal logic, many "possible worlds" are considered. A truth is necessary if it is true in all possible worlds. By contrast, if a statement happens to be true in our world, but is false in another world, then it is a contingent truth. A statement that is true in some world (not necessarily our own) is called a possible truth.
Furthermore, the proof uses higher-order (modal) logic because the definition of God employs an explicit quantification over properties.
First, Gödel axiomatizes the notion of a "positive property": for each property φ, either φ or its negation ¬φ must be positive, but not both (axiom 2). If a positive property φ implies a property ψ in each possible world, then ψ is positive, too (axiom 1).[note 3] Gödel then argues that each positive property is "possibly exemplified", i.e. applies at least to some object in some world (theorem 1). Defining an object to be Godlike if it has all positive properties (definition 1), and requiring that property to be positive itself (axiom 3), Gödel shows that in some possible world a Godlike object exists (theorem 2), called "God" in the following. Gödel proceeds to prove that a Godlike object exists in every possible world.
To this end, he defines essences: if x is an object in some world, then a property φ is said to be an essence of x if φ(x) is true in that world and if φ necessarily entails all other properties that x has in that world (definition 2). Requiring positive properties being positive in every possible world (axiom 4), Gödel can show that Godlikeness is an essence of a Godlike object (theorem 3). Now, x is said to exist necessarily if, for every essence φ of x, there is an element y with property φ in every possible world (definition 3). Axiom 5 requires necessary existence to be a positive property.
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