# Gödel's ontological proof

Gödel's ontological proof is a formal argument for God's existence by the mathematician Kurt Gödel (1906–1978).

It 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

4. There are other worlds and rational beings of a different and higher kind.
5. The world in which we live is not the only one in which we shall live or have lived.
13. There is a scientific (exact) philosophy and theology, which deals with concepts of the highest abstractness; and this is also most highly fruitful for science.
14. Religions are, for the most part, bad—but religion is not.[citation needed]

## History of Gödel's proof

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)."[1] 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.[2]

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,[3] Gödel argued at length for a belief in an afterlife.[4] 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."[5] 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."[6] 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."[7]

## Outline of Gödel's proof

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 the planet is covered by water". 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.

From axioms 1 through 4, Gödel argued that in some possible world there exists God. He used a sort of modal plenitude principle to argue this from the logical consistency of Godlikeness. Note that this property is itself positive, since it is the conjunction of the (infinitely many) positive properties.

Then, Gödel defined essences: if x is an object in some world, then the property P is said to be an essence of x if P(x) is true in that world and if P entails all other properties that x has in that world. We also say that x necessarily exists if for every essence P the following is true: in every possible world, there is an element y with P(y).

Since necessary existence is positive, it must follow from Godlikeness. Moreover, Godlikeness is an essence of God, since it entails all positive properties, and any nonpositive property is the negation of some positive property, so God cannot have any nonpositive properties. 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.

From these hypotheses, it is also possible to prove that there is only one God in each world by Leibniz's law, the identity of indiscernibles: two or more objects are identical (are one and the same) if they have all their properties in common, and so, there would only be one object in each world that possesses property G. Gödel did not attempt to do so however, as he purposely limited his proof to the issue of existence, rather than uniqueness. This was more to preserve the logical precision of the argument than due to a penchant for polytheism. This uniqueness proof will only work if one supposes that the positiveness of a property is independent of the object to which it is applied, a claim which some have considered to be suspect[who?].

To formalize the argument sketched above, the following definitions and axioms are needed:

• Definition 1: x is God-like if and only if x has as essential properties those and only those properties which are positive
• Definition 2: A is an essence of x if and only if for every property B, x has B necessarily if and only if A entails B
• Definition 3: x necessarily exists if and only if every essence of x is necessarily exemplified
• Axiom 1: Any property entailed by—i.e., strictly implied by—a positive property is positive
• Axiom 2: A property is positive if and only if its negation is not positive
• Axiom 3: The property of being God-like is positive
• Axiom 4: If a property is positive, then it is necessarily positive
• Axiom 5: Necessary existence is a positive property

Axiom 1 assumes that it is possible to single out positive properties from among all properties. Gödel comments that "Positive means positive in the moral aesthetic sense (independently of the accidental structure of the world)... It may also mean pure attribution as opposed to privation (or containing privation)." (Gödel 1995). Axioms 2, 3 and 4 can be summarized by saying that positive properties form a principal ultrafilter.

From these axioms and definitions and a few other axioms from modal logic, the following theorems can be proved:

• Theorem 1: If a property is positive, then it is consistent, i.e., possibly exemplified.
• Theorem 2: The property of being God-like is consistent.
• Theorem 3: If something is God-like, then the property of being God-like is an essence of that thing.
• Theorem 4: Necessarily, the property of being God-like is exemplified.

Symbolically:

$\begin{array}{rl} \text{Ax. 1.} & \left\{P(\varphi) \wedge \Box \; \forall x[\varphi(x) \to \psi(x)]\right\} \to P(\psi) \\ \text{Ax. 2.} & P(\neg \varphi) \leftrightarrow \neg P(\varphi) \\ \text{Th. 1.} & P(\varphi) \to \Diamond \; \exists x[\varphi(x)] \\ \text{Df. 1.} & G(x) \iff \forall \varphi [P(\varphi) \to \varphi(x)] \\ \text{Ax. 3.} & P(G) \\ \text{Th. 2.} & \Diamond \; \exists x \; G(x) \\ \text{Df. 2.} & \varphi \text{ ess } x \iff \varphi(x) \wedge \forall \psi \left\{\psi(x) \to \Box \; \forall y[\varphi(y) \to \psi(y)]\right\} \\ \text{Ax. 4.} & P(\varphi) \to \Box \; P(\varphi) \\ \text{Th. 3.} & G(x) \to G \text{ ess } x \\ \text{Df. 3.} & E(x) \iff \forall \varphi[\varphi \text{ ess } x \to \Box \; \exists y \; \varphi(y)] \\ \text{Ax. 5.} & P(E) \\ \text{Th. 4.} & \Box \; \exists x \; G(x) \end{array}$

There is an ongoing open-source effort to formalize Gödel's proof using various theorem provers and proof assistants. The formalized proof of God's existence made headlines in German newspapers.[8]