Sorry. Wrongo.
Godel's proof of God
P(psi) P is "positive"
G(x) x have the property God
ess. essential
E existing
• (bullet) Necessary
This "proof" on the existence of God, hasn't been published. But Dana Scott who belonged to the closest circle around Gödel, claimed this were constructed by Gödel. In 1970 Dana Scott sent this alledged proof to Stig Kanger. Stig Kanger was a professor of theoretical philosophy in Uppsala, but died in 1988. I was one of his latest pupils. Stig distributed this proof at the department in Uppsala and perhaps it also has been distributed in the USA.
From axioms 1 through 4, Godel 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(x) 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 identity of indiscernibles, no two distinct objects can have precisely the same properties, and so there can 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.
The Synthetic Proof
A Synthetic Proof that God Exists
The following proof uses Godels' theorem as a stepping stone to proving the existence of an infinite mind, which is then identified with God.
The proof is in two steps.
Step One: Existence Proof of Higher Worlds
Let us suppose that human reason forms a closed system. By 'human reason' I mean the set of synthetic a priori principles that delineate the categories of human thought, together with the principles of classical logic. Godel's incompleteness proof showed that any closed system is incomplete in the sense that there are true sentences that are unprovable in the system. These sentences are sentences about the system itself, such as "S is consistent", where 'S' refers to some closed system. These sentences can only be proved by moving up a type into a higher order "metasystem". Now consider the sentence "S is consistent" as applied to human reason. It follows from Godel's incompleteness proof that this sentence is unprovable in the system of human reason.
Nevertheless, we do have inductive, empirical, and pragmatic grounds for believing that human reason is consistent. We have an inductive basis in that other smaller systems of a lower order can be proved to be consistent by means of a higher order system. We have empirical grounds in that we have yet to deduce a contradiction from the laws of classical logic. We have a pragmatic basis in that the use of human reason has proved to be of practical value as a framework for conceiving the world.
But if the sentence "S is consistent", as applied to human reason, is true, it is provable, as is shown by an application of Leibniz's Principle of Sufficient Reason. This principle states that there is a sufficient reason for everything. Leibniz intended this to mean that there is an a priori proof for every true sentence (or proposition).
But a proof of "S is consistent" cannot be given in the system of human reason (our system). In order to do this, one would have to transcend the bound of human reason, which, as we all know, cannot be done. In less picturesque terms, the proof could only be given in a system more powerful than ours, one which is "up a type" from ours.
But it follows from the incompleteness proof and Leibniz's Law that there must be such a system. Hence, higher worlds exist.
Step Two: A Synthetic Proof that God Exists
Since the laws of logic describe the way our minds necessarily work, we cannot conceive of what a Super Logic of the kind whose existence has been proved is like. We stand with respect to such a system in the same way a person who is a point on a line stands with respect to flatland, or in the same way in which a person in flatland stands with respect to a three-dimensional world. We are simply unable to conceive of such a world. Nevertheless, such a world must exist, as has been demonstrated.
Because no logical system can exist apart from some mind, the existence of a Super Logic requires the existence of a higher mind.
Since there are infinitely many such logical systems (the same argument could be repeated for each logical system), these systems must describe the workings of an infinite mind. But only God can have an infinite mind. Therefore, God exists. |
