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Gödel's ontological proof is a formalization of Saint Anselm's ontological argument for God's existence by the mathematician Kurt Gödel.
St. Anselm's ontological argument, in its most succinct form, is as follows: "God, by definition, is that than which a greater cannot be thought. 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; this is the version that Gödel studied and attempted to clarify with his ontological argument.
While Gödel was religious, he never published his proof because he feared that it would be mistaken as establishing God's existence beyond doubt. Instead, he only saw it as a logical investigation and a clean formulation of Leibniz' argument with all assumptions spelled out. He repeatedly showed the argument to friends around 1970 and it was published after his death in 1987. An outline of the proof follows.
Contents 
Modal logic
The proof uses modal logic, which distinguishes between necessary truths and contingent truths.
A truth is necessary if it cannot be avoided, such as 2 + 2 = 4; by contrast, a contingent truth just happens to be the case, for instance "more than half of the earth is covered by water". In the most common interpretation of modal logic, one considers "all possible worlds". If a statement is true in all possible worlds, then it is a necessary truth. If a statement happens to be true in our world, but is not true in all other worlds, then it is a contingent truth. A statement that is true in some world (not necessarily our own) is called a possible truth.
A property assigns to each object in every possible world a truth value (either true or false). Note that not all worlds have the same objects: some objects exist in some worlds and not in others. A property has only to assign truth values to those objects that exist in a particular world. As an example, consider the property
 P(x) = x is grey
and consider the object
 s = my shirt
In our world, P(s) is true because my shirt happens to be grey; in some other world, P(s) is false, while in still some other world, P(s) wouldn't make sense because my shirt doesn't exist there.
We say that the property P entails the property Q, if any object in any world that has the property P in that world also has the property Q in that same world. For example, the property
 P(x) = x is taller than 2 meters
entails the property
 Q(x) = x is taller than 1 meter.
Axioms
We first assume the following axiom:
 Axiom 1: It is possible to single out positive properties from among all properties. Gödel defines a positive property rather vaguely: "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)
We then assume that the following three conditions hold for all positive properties (which can be summarized by saying "the positive properties form an ultrafilter"):
 Axiom 2: If P is positive and P entails Q, then Q is positive.
 Axiom 3: If P_{1}, P_{2}, P_{3}, ..., P_{n} are positive properties, then the property (P_{1} AND P_{2} AND P_{3} ... AND P_{n}) is positive as well.
 Axiom 4: If P is a property, then either P or its negation is positive, but not both.
Finally, we assume:
 Axiom 5: Necessary existence is a positive property (Pos(NE)). This mirrors the key assumption in Anselm's argument.
Now we define a new property G: if x is an object in some possible world, then G(x) is true if and only if P(x) is true in that same world for all positive properties P. G is called the "Godlike" property. An object x that has the Godlike property is called God.
Derivation
Subject to the assumptions, it is asserted that one can now already show that in some possible world there exists God. We want to show that necessarily, in every world there exists a unique God.
In order to do this, Gödel first defines 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 strongly exists if for every essence P(x) the following is true: in every possible world, there is an element y with P(y).
From these hypotheses, it is now possible to prove that there is one and only one God in each world (crudely, no two distinct objects can have precisely the same properties, for example one property that x has is is x, and hence the property G is unique, only one object in a possible world can possess it). 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.
Critique of definitions and axioms
There are several reasons Gödel's axioms may not be realistic, including the following:
 It may be impossible to satisfy the second axiom. Since it could not be proven consistent, this axiom was replaced in some reworkings of the proof by the assumption that G(x) is positive (Pos(G(x)).
 Axiom 3 assumes that a conjunction of positive properties is also a positive property, but some positive properties may be incompatible with others. For example mercy may be incompatible with justice. In that case the conjunction would be an impossible property and G(x) would be false of every x. Ted Drange has made this objection to the coherence of attributing all positive properties to God. See http://www.infidels.org/library/modern/theodore_drange/incompatible.html for Drange's list of incompatible properties and some counter arguments.
 It was pointed out by Jordan Sobel that Gödel's axioms are too strong: they imply that all possible worlds are identical. C. Anthony Anderson gave a slightly different axiomatic system which attempts to avoid this problem.
See also
 Absolute Infinite
 Arguments for the existence of God
 Modality
 Philosophy of religion
 Mathematics and God
 Synthetic proposition
 Teleological argument
 Theism
External links
 Kurt Gödel's Ontological Argument (http://www.stats.uwaterloo.ca/~cgsmall/ontology.html)
 Stanford Encyclopedia of Philosophy: Gödel's Ontological Argument (http://plato.stanford.edu/entries/ontologicalarguments/#6)
References
 C. Anthony Anderson, "Some Emendations of Gödel's Ontological Proof", Faith and Philosophy, Vol. 7, No 3, pp. 291303, July 1990
 Kurt Gödel (1995). "Ontological Proof". Collected Works: Unpublished Essays & Lectures, Volume III. pp. 403404. Oxford University Press. ISBN 0195147227
 A. P. Hazen, "On Gödel's Ontological Proof", Australasian Journal of Philosophy, Vol. 76, No 3, pp. 361377, September 1998
 Jordan Howard Sobel, "Gödel's Ontological Proof" in On Being and Saying. Essays for Richard Cartwright, ed. Judith Jarvis Thomson (MIT press, 1987)