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Gödel's ontological proof

Gödel's ontological proof is a formalization of St. 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 none is greater. 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.

Table of contents
1 Modal logic
2 Axioms
3 Critique of Definitions and Axioms
4 Derivation
5 Related Articles
6 Related Articles (Objections)
7 External Links
8 References

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 only has 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.


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 P1, P2, P3, ... are positive properties, then the property (P1 AND P2 AND P3 ...) 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 "God-like" property. An object x that has the God-like property is called God.

Critique of Definitions and Axioms

There are several reasons these assumptions are not realistic, including the following:

Since the final axiom is Anselm's, it is discussed under ontological argument.


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 of 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.

[Some explanation of how, rather than "it is now possible to do so" would make this article more useful.]

It was pointed out by Sobel that Gödel's axioms are too strong: they imply that all possible worlds are identical. Anderson gave a slightly different axiomatic system which attempts to avoid this problem.

Related Articles

Absolute Infinite, Arguments for the existence of God, Modality, Philosophy of religion, Synthetic proposition, Teleological argument, Theism

Related Articles (Objections)

Arguments against the existence of God, Logical and evidential arguments from evil, The problem of evil

External Links