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.
Gödel left a fourteen-point outline of his philosophical beliefs in his papers.
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.)
[3] In letters to his mother, who was not a churchgoer and had raised Kurt and his brother as freethinkers,[4] Gödel argued at length for a belief in an afterlife.
[5] 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.
"[7] In an unmailed answer to a questionnaire, Gödel described his religion as "baptized Lutheran (but not member of any religious congregation).
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.
[note 3] 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),[note 4] and requiring that property to be positive itself (axiom 3),[note 5] Gödel shows that in some possible world a Godlike object exists (theorem 2), called "God" in the following.
[note 6] Gödel proceeds to prove that a Godlike object exists in every possible world.
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 (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.
Common notation in symbolic logic: Modal operators (used in modal logic): Primitive predicate in this argument: Derived predicates (defined in terms of other predicates within the argument): Possible-worlds readings of modal terms have been added in parentheses, i.e., "in all possible worlds" for "necessarily" and "in at least one possible world" for "possibly".
For completeness, "in the actual world" should be added to all sentences that were said without "necessarily" or "possibly", but this has been skipped since it might make the text difficult to read.
It is particularly applicable to Gödel's proof – because it rests on five axioms, some of which are considered questionable.
The first layer of criticism is simply that there are no arguments presented that give reasons why the axioms are true.
This line of thought was argued by Jordan Howard Sobel,[12] showing that if the axioms are accepted, they lead to a "modal collapse" where every statement that is true is necessarily true, i.e. the sets of necessary, of contingent, and of possible truths all coincide (provided there are accessible worlds at all).
[note 7] According to Robert Koons,[9]: 9 Sobel suggested in a 2005 conference paper[citation needed] that Gödel might have welcomed modal collapse.
Robert J. Spitzer accepted Gödel's proof, calling it "an improvement over the Anselmian Ontological Argument (which does not work).
"[18] There are, however, many more criticisms, most of them focusing on the question of whether these axioms must be rejected to avoid odd conclusions.
The broader criticism is that even if the axioms cannot be shown to be false, that does not mean that they are true.
Hilbert's famous remark about interchangeability of the primitives' names applies to those in Gödel's ontological axioms ("positive", "god-like", "essence") as well as to those in Hilbert's geometry axioms ("point", "line", "plane").
According to André Fuhrmann (2005) it remains to show that the dazzling notion prescribed by traditions and often believed to be essentially mysterious satisfies Gödel's axioms.
In the same paper, they suspected Gödel's original version of the axioms[note 12] to be inconsistent, as they failed to prove their consistency.
[note 13] In 2016, they gave an automated proof that the original version implies
[24]: 940 lf Moreover, they gave an argument that this version is inconsistent in every logic at all,[note 14] but failed to duplicate it by automated provers.
[note 15] However, they were able to verify Melvin Fitting's reformulation of the argument and guarantee its consistency.
[25] A humorous variant of Gödel's ontological proof is mentioned in Quentin Canterel's novel The Jolly Coroner.
[26][page needed] The proof is also mentioned in the TV series Hand of God.