Published on December 8, 2025 8:49 PM GMT

In 1970, Gödel — amidst the throes of his worsening hypochondria and paranoia —entrusted to his colleague Dana Scott^[1] a 12-line proof that he had kept mostly secret since the early 1940s. He had only ever discussed the proof informally in hushed tones among the corridors of Princeton’s Institute for Advanced Study, and only ever with close friends: Morgenstern, and likely Einstein^[2].

This proof purported to demonstrate that there exists an entirely good God. This proof went unpublished for 30 years due to Gödel’s fears of being seen as a crank by his mathematical colleagues — a reasonable fear during the anti-metaphysical atmosphere that was pervading mathematics and analytic philosophy at the time, in the wake of the positivists.

Oskar Morgenstern remarked:

Über sein ontologischen Beweis — er hatte das Resultat vor einigen Jahren, ist jetzt zufrieden damit aber zögert mit der Publikation. Es würde ihm zugeschrieben werden daß er wirkl[ich] an Gott glaubt, wo er doch nur eine logische Untersuchung mache (d.h. zeigt, daß ein solcher Beweis mit klassischen Annahmen (Vollkommenheit usw.), entsprechend axiomatisiert, möglich sei)’

Oskar Morgenstern

About his ontological proof — he had the result a few years ago, is now happy with it but hesitates with its publication. It would be ascribed to him that he really believes in God, where rather he is only making a logical investigation (i.e. showing such a proof is possible with classical assumptions (perfection, etc.) appropriately axiomatized.

Oskar Morgenstern, translation (mine)

Gödel ultimately died in 1978, and the proof continued to circulate informally for about a decade. It was not until 1987, when the collected works of Gödel were released, that the proof was published openly. Logicians perked up at this, as when Gödel — who, it’s fair to say, is one of the all-time logic GOATs^[3]— says that he has found a logical proof of the existence of God you would do well to consider it seriously indeed.

The Proof

The Logic

Godel’s proof takes place in a canonical version of modal logic. This form of modal logic is usually called S5. It contains all the standard rules of propositional logic: modus ponens, conjunction; all of the standard tautologies. As well as four rules for the so-called “modal operators.” The modal operators are represented by a “Box” and a “Diamond”. Whenever you see a box, you should think “it is necessary that…”, or “in all possible worlds” and whenever you see a diamond, you should read “it is possible that…”, or “in at least one possible world.”

Rule 1

Brief justification. This one is pretty easy. It says only that if it is necessary that A is true (i.e. there is no world where A is false), then A must also just actually be true. We won’t really need to use this rule.

Rule 2

Brief justification. This rule states that if it is necessary that “A implies B,” then if it is necessary that A is true, it is also necessary that B is true. You can imagine this as saying the following:

1. Suppose that there are no worlds where A is true and B is not true (so that it is necessary that A implies B).
2. Then also, if in every world A is true.
3. Then in every world B is also true.

Rule 3

Brief justification. This rule says that if it is possible for A to be true, then it is necessary that A is possibly true. That is, if in some world it is possible for A to have been true (so that A is contingent), then it is also necessary that A could have been true, so that A could have been true in any world.

This is the most contentious rule of S5, but it is equivalent to some other rules that are less contentious — I think it’s not so bad. But you already know the final stop on this train — you may choose to get off here, though I would recommend a later station.

Rule 4 - Necessitation rule

Brief Justification. This rule just states that if a logical statement follows from nothing, i.e. it is a logical tautology, then it is necessarily true in all possible worlds. This makes sense, since we suppose that the rules of logic apply in all possible worlds (the possible worlds we’re considering here are supposed to represent all the logically possible worlds). There is no logically possible world where a tautology is not true, since it is a tautology (a note that this statement is itself basically a tautology).

This rule is usually not stated explicitly as one of the axioms of S5, as it is so widely accepted that all of the modal logics basically take it as given. The modal logics that satisfy this rule are called the normal modal logics.

The Proof - Axioms and Definitions

I will walk through the proof that Scott transcribed^[4], which is in the usual logical notation that modern philosophers and mathematicians are familiar with. It is equivalent to Gödel’s original proof, but Gödel wrote his version in traditional logic notation — which is in my opinion much harder to read. The proof takes place in second-order logic, so we will have “properties of properties” (i.e. a property can be “perfect^[5]” or a property can be “imperfect”). This is not particularly controversial.

Axiom 1 - Monotonicity of perfection

I know it’s starting to look scary — but I promise it’s not as bad as it looks. The P that appears here is one of those “properties of properties” I mentioned just before. It says only that “this property is perfect”.

The sentence above says, therefore, that “If property 1 is perfect, and it it is necessarily true that whenever an object has property 1 it must also have property 2, then property 2 must also be perfect.”

This seems justifiable enough — if property 1 is perfect, and there is no world in which you have property 1 and don’t also have property 2, then property 2 should also be perfect, otherwise how could we have said that property 1 was truly perfect in the first place?

Axiom 2 - Polarity of perfection

This says that if a property is perfect, then not having that property is not perfect. This just means that it can’t be the case that having a certain property is perfect and also not having that property is perfect. This seems sensible enough — we cannot say it is perfect to be all-knowing and also perfect to not be all-knowing, this would be nonsensical.

Notably, this is not saying that every property is either perfect or not perfect — it is just saying that if a property is perfect, then the negation of that property can’t also be perfect.

Axiom 3 - Possibility of perfection

This axiom says that if a property is perfect, then it must be possible for there to exist something with that property.

It would seem unreasonable to say that a property is perfect and also that there are no worlds where anything can actually instantiate that property. Surely it must at least be possible for something to have the property if we’re calling it perfect, even if nothing in our world actually has that property. Otherwise we can just resign this property to the collection of neither-perfect-nor-imperfect properties, or else simply call it imperfect, since nothing can ever have it.

Definition 1 - Definition of God

This introduces the definition of God that we’ll be working with in this proof. It should be relatively familiar — something has the property of “Being God” if it possesses every perfect property. Also, if something possesses every perfect property, then we can call that thing God. It is a being which possesses every perfect property, what word would you like to use for it?

It does not say that God only possesses perfect properties, God can also possess properties that are neither perfect nor imperfect — however God cannot have any imperfect properties, as that would lead to a contradiction by Axiom 2.

Axiom 4 - God is Perfect

This axiom just says that the property of being God — you know, the property of having every perfect property — is itself perfect.

This seems reasonable, it almost follows from axiom 1, however since there is no one perfect property that implies you are God, God is not perfect as a result of axiom 1, so we need to introduce a special axiom to say that God is perfect.

Again, how would you describe a being that has every perfect property — it seems ridiculous to say that such a being is not itself perfect.

Definition 2 - Perfect-Generating Property