Gödel's ontological proof

Gödel's ontological proof is a formal argument by the mathematician Kurt Gödel (1906–1978) for the existence of God. The argument is in a line of development that goes back to Anselm of Canterbury (1033–1109). St. Anselm's ontological argument, in its most succinct form, is as follows: 'God, by definition, is that for which no greater can be conceived. God exists in the understanding. If God exists in the understanding, we could imagine Him to be greater by existing in reality. Therefore, God must exist.' A more elaborate version was given by Gottfried Leibniz (1646–1716); this is the version that Gödel studied and attempted to clarify with his ontological argument. Gödel's ontological proof is a formal argument by the mathematician Kurt Gödel (1906–1978) for the existence of God. The argument is in a line of development that goes back to Anselm of Canterbury (1033–1109). St. Anselm's ontological argument, in its most succinct form, is as follows: 'God, by definition, is that for which no greater can be conceived. God exists in the understanding. If God exists in the understanding, we could imagine Him to be greater by existing in reality. Therefore, God must exist.' A more elaborate version was given by Gottfried Leibniz (1646–1716); this is the version that Gödel studied and attempted to clarify with his ontological argument. Gödel left a fourteen-point outline of his philosophical beliefs in his papers. Points relevant to the ontological proof include The first version of the ontological proof in Gödel's papers is dated 'around 1941'. Gödel is not known to have told anyone about his work on the proof until 1970, when he thought he was dying. In February, he allowed Dana Scott to copy out a version of the proof, which circulated privately. In August 1970, Gödel told Oskar Morgenstern that he was 'satisfied' with the proof, but Morgenstern recorded in his diary entry for 29 August 1970, that Gödel would not publish because he was afraid that others might think 'that he actually believes in God, whereas he is only engaged in a logical investigation (that is, in showing that such a proof with classical assumptions (completeness, etc.) correspondingly axiomatized, is possible).' Gödel died January 14, 1978. Another version, slightly different from Scott's, was found in his papers. It was finally published, together with Scott's version, in 1987. Morgenstern's diary is an important and usually reliable source for Gödel's later years, but the implication of the August 1970 diary entry—that Gödel did not believe in God—is not consistent with the other evidence. In letters to his mother, who was not a churchgoer and had raised Kurt and his brother as freethinkers, Gödel argued at length for a belief in an afterlife. He did the same in an interview with a skeptical Hao Wang, who said: 'I expressed my doubts as G spoke Gödel smiled as he replied to my questions, obviously aware that his answers were not convincing me.' Wang reports that Gödel's wife, Adele, two days after Gödel's death, told Wang that 'Gödel, although he did not go to church, was religious and read the Bible in bed every Sunday morning.' In an unmailed answer to a questionnaire, Gödel described his religion as 'baptized Lutheran (but not member of any religious congregation). My belief is theistic, not pantheistic, following Leibniz rather than Spinoza.' 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, a truth is contingent if it just happens to be the case. For instance, 'more than half of this planet is covered by water' is a contingent truth, that relies upon which planet 'this planet' is. 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). 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. Hence, it must follow from Godlikeness. Moreover, Godlikeness is an essence of God, since it entails all positive properties, and any non-positive property is the negation of some positive property, so God cannot have any non-positive properties. Since necessary existence is also a positive property (axiom 5), it must be a property of every Godlike object, as every Godlike object has all the positive properties (definition 1). Since any Godlike object is necessarily existent, it follows that any Godlike object in one world is a Godlike object in all worlds, by the definition of necessary existence. Given the existence of a Godlike object in one world, proven above, we may conclude that there is a Godlike object in every possible world, as required (theorem 4). Besides axiom 1-5 and definition 1-3, a few other axioms from modal logic were tacitly used in the proof.