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 This proof went unpublished for 30 years due to Gödel's fears of being seen as a crank by his mathematical colleagues — a reasonable fear during the anti-metaphysical atmosphere that was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

This definition introduces the notion of an “Perfect-Generating” property. This basically says that a perfect-generating property of a thing is a property that captures everything that makes that thing “perfect.”^[6]

For example, suppose that the perfect properties of a triangle are things like:

• having all sides equal
• having all angles equal
• being maximally symmetric
• having maximal geometric regularity

Then the perfect-generating property of a triangle would be the property of “being an equilateral triangle,” as that implies all of the other perfect properties.

To describe the logic of the statement more explicitly, it says:

We say property 1 is a perfect-generating property of X iff X has property 1, and for any other perfect property 2, if X has property 2, then anything else that has property 1 must necessarily (i.e. in all possible worlds) also have property 2.

So it cannot be the case that there is some Y that has the perfect-generating property of X, and yet X has some perfect property that Y does not have. In fact, there is no world where that happens — it is necessary that anything with this perfect-generating property also has all the other perfect properties of anything else with that same perfect-generating property.

Note that this doesn’t mean that the “perfect-generating” property is itself a perfect property (although it will be whenever we use it in the proof).

Definition 3 - Perfect-Essential Necessity

This is just another definition.

We say that something has the property of “perfect-essential necessity” if (and only if) whenever there is a perfect-generating property for that thing, it implies that there necessarily must exist something (i.e. in every world) which instantiates that perfect-generating property.

This would also mean that there must necessarily exist a thing that has all the perfect properties that follow from that perfect-generating property.

This is just a definition of what it would mean for something to have “perfect-essential necessity.” We have not claimed that any such thing exists, we are just saying that if a thing has a perfect-generating property that must necessarily exist, then it also has the property of “perfect-essential necessity.”

It is just a definition — it cannot hurt you.

Axiom 5 - Necessary Perfection

This axiom states that whenever we say that property is perfect, it must be perfect in all possible worlds. It must be necessary that the property is perfect.

If there is some world in which a property is not perfect, this axiom claims, then it was never truly perfect to begin with. We are not talking about your “garden-variety” perfection — we’re only talking about the real cream of the crop perfect properties.

Axiom 6 - Perfect-Essential Necessity is Perfect

This claims that the property of having “perfect-essential necessity” — whereby all your perfect properties must necessarily be instantiated in the world — is itself a perfect property. This seems sensible to me! If it were the case that all of the perfect properties of a thing had to be instantiated in the world, it seems like having that property would also be perfect.

Is it not itself a perfect property for all of your perfect properties to be instantiated in the world? Is it somehow imperfect for all of your perfect properties to be required to exist — well then clearly they were not perfect to begin with!

The Proof - Derivation

Well, if you haven’t gotten off the logic-train at any of the steps above, then you know what’s coming now. Let’s show that these axioms entail the existence of God.

Theorem 1

This says “it is possible for there to exist an X, such that X is God.”

We prove this using axiom 3, which stated that any perfect property must have the property that it is possible that something instantiates that perfect property.

Then we apply this to axiom 4, which said that “Being God” is a perfect property.

So, just by replacing the φ in axiom 3 with G, it follows that it must be possible for there to exist a God.

Theorem 2

This says that “If there is an X with the property of Being God, then Being God is a perfect-generating property of that X,” i.e. it captures all the other perfect properties that such an X would have.”

This is a more involved derivation, so let’s go step by step. What we need to show is that if there is something with the property of “Being God”, then it satisfies the definition of a perfect-generating property. So let’s look at definition 2 more closely.

So we need to show the conjunction on the right holds when φ is replaced with G. It is clear that the first clause of the conjunction holds, i.e. if something has the property of “Being God,” then it has the property of “Being God.” That was easy!

The trickier part is showing the second clause. This says that for any property ψ, if God has that property and that property is positive, then it is necessarily the case that whenever anything has the property of “Being God,” it must also have that positive property ψ.

Let’s begin by again noting the definition of God

And since this is a definition, we can apply the rule of universal generalization from logic^[7]. We’ll also drop the reverse implication, since it isn’t necessary for the result.

This is basically just a restatement of our definition. It’s saying for anything which has the property of “Being God,” it also satisfies our definition of “Being God.” We’ve also replaced φ with ψ, just so it eventually lines up with our definition of a perfect-generating property — but we could’ve replaced it with anything.

However, since this is a theorem that we can derive from just our definition of God and the rules of logic alone, we can apply rule 4 from our modal logic system, the necessitation rule. This must apply in any world (since we assume that it is necessarily true that all the laws of logic are true). Therefore, we can just write that the above statement is necessary.

Now we have that in every possible world, “Being God” implies that if a property is positive, you have that property.

From this point on, it’s going to get a little rocky if you aren’t familiar with basic logic. Explaining the rules of basic logic would unfortunately take too long, and this post is long enough already, but feel free to skip to Theorem 3. I promise nothing here is a trick, or at all contentious.

To reach the conclusion for the definition of a perfect-generating property, we can begin by assuming that the ψ we’re working with is perfect. Otherwise there’s nothing we need to show about it (since the first clause of the implication we’re trying to show would be false, so the implication would be vacuously true). So if we assume this, axiom 5 lets us say:

And therefore we can also assume that it is necessary that ψ is a perfect property. Then we can apply a theorem of S5’s modal logic (a basic corollary of proposition 3.17 here, but don’t worry about it too much), to Lemma 1 and get:

Then we can generalize back again, going back to thinking about this being true for anything we include in the formula — since we had a general formula originally in Lemma 1 — which gives us:

Which is the conclusion that we wanted to reach on the right hand side of the definition of a perfect-generating property. So we’ve shown Theorem 2 must hold.

Theorem 3

This is the final theorem we’ll need to prove. That it is necessary for there to exist a being with the property of “Being God.”

For this, we’ll need to prove a small lemma, which is:

To prove this lemma, we begin by assuming there exists something that has the property of “Being God;” from this we then want to show the right-hand side, that there must then necessarily exist something with this property. Let’s pick an arbitrary thing that has this property of “Being God” to work with^[8], so that we have:

for some a. Then by axiom 6 — that perfect-essential necessity is itself a perfect property, and so it is a property that anything with the property of “Being God” must have, were such a thing to exist — and the definition of God, we have:

Then, by restating theorem 2 for this generic a that we’ve chosen:

Then we use the definition of perfect-essential necessity, and plug in “Being God” as the perfect-generating property (we only need to use the forward implication from the definition):

So, finally, let’s combine our assumption that there exists something with the property of being God, with statements (1), (2), and (3). Then following the chain of implications, we arrive at:

So it is necessary that there exists something with the property of “Being God.” But remember, we have to include the initial assumption we used to prove this, so ultimately all we have shown is:

So the lemma has been proven.

Then we need one last step from our modal logic, we’ll start with theorem 1, which stated:

Then we’ll apply our Lemma 2, to change existence into necessary existence, to get:

So we have that it is possible for it to be necessary for there to exists something that has the property of “Being God.” Then finally, we’ll apply Rule 3^[9] from our modal logic to get:

And so it is necessarily true that there exists something with the property of “Being God.” That is, there is something that is “Being God” in every possible world.

Checkmate, atheists.

Conclusion

Well, that was a pretty intense modal-logic session we just went through. I hope that I managed to walk you through it effectively, and perhaps hope that I managed to convince you that there is a God — though I must say, I have my doubts.

Now, I departed from the proof in one place, which was my statement of Axiom 2. Originally, Gödel stated it so that every property is either perfect or it’s negation is perfect, and I have stated it to leave open the possibility that some properties can be neither perfect nor imperfect. This also meant that I had to modify the definition of a perfect-generating property, which can be phrased in a slightly cleaner way if you accept that every property is perfect or imperfect — but I prefer my phrasing^[10]. The proof goes through the same way. It is worth noting, though, that in Gödel’s original proof, the bipartite nature of perfect properties (i.e. that every property is perfect or imperfect) ends up leading to something called “modal collapse,” in which everything that is true is also necessarily true — which seems like an unfortunate consequence. I am not certain that my formulation totally avoids modal collapse — though it certainly prevents the most obvious way to get there.

However, I remain uncertain about the existence of God — why is that? After the proof I just gave. Well, it is because I am not sure that any property is perfect, aside from perhaps the property of “Being God”, and the property of “perfect-essential necessity.” I think my arguments for those being perfect properties are good enough, though is it the correct turn-of-phrase to say something has the property of “Being God” if it only has that property, and every other property is simply contingent — I am not so sure. There also remain ontological uncertainties about whether S5 is a legitimate modal logic to consider for our reality. Rule 3 is fairly strong, and I am not sure if I want to accept it, and as you saw in Theorem 3, that was a fairly essential part of the proof — the whole edifice would collapse if we rejected that. But it is implied by some less-controversial axioms, as I mentioned, so perhaps we ought to believe it.

One other comment on the proof — as with so many ontological arguments, we could go through the proof and entirely modify the language, so that instead we are talking about “Perfectly Evil” properties, or indeed any other properties^[11]. Perhaps, what this proof has really done is shown us the truth of Manichaeism.

A note that this proof has actually been formalized (in Gödel’s original formulation, not my variant) in a computer-assisted theorem prover! I’m sure the Germans who did that realized the magnitude of the headlines they could generate — “Proof of God Verified by Computers!” Perhaps those of us engaged in similarly wacky disputes could formalize an argument (that works, as all arguments do, only by accepting certain axioms) and then generate a headline that says what you actually believed all along has now been demonstrated beyond any doubt to be certain. This may be a fruitful generalizable PR stratagem.

1. ^{^}

   Unfortunately, I’m talking about Dana Scott the logician — of Scott’s Trick fame — not the incredibly attractive lawyer from world-renowned TV show Suits.

2. ^{^}

   There is no evidence for Gödel sharing it with Einstein, though it is well-known that Gödel and Einstein were incredibly close during their time at Princeton, so it is hard to imagine Gödel never brought it up with Einstein.

3. ^{^}

   I know that means all-time logic greatests of all time, there’s two all-times. I think Gödel probably deserves two all-time awards.

4. ^{^}

   Okay, I’ve made a minor modification to make it more satisfying to me personally. I’ll discuss this in the conclusion.

5. ^{^}

   I will use perfect to mean the classical English definition of perfect, not the technical Leibnizian meaning. The argument feels more justifiable to me in this vocabulary than the standard vocabulary of “positive” or “good.”

6. ^{^}

   This is where I’ve made a slight departure from Gödel’s original proof — I think the proof I’m going to provide here is just a bit more convincing, since otherwise you need to have a biconditional on perfect properties (i.e. every property is perfect or its negation is perfect), and the essential property needs to capture everything about something, not just everything that is perfect about it. There are reasons to prefer Gödel’s original formulation, but I prefer mine (of course). Don’t worry, I’ll talk about the original and my twist on it in the conclusion.

7. ^{^}

   I promise we’re allowed to do this.

8. ^{^}

   Again, I promise this is okay.

    

9. ^{^}

   This isn’t quite rule 3, but it’s a corollary of Rule 3. I’ll write a quick proof here. What we want is:

   $$◊\square p\to \square p$$

   and what we have is:

   $$◊p\to \square ◊p$$

   We get this through a process called “taking the dual.” Begin by replacing p with ~p:

   $$◊\neg p\to \square ◊\neg p$$

   Then contrapose to get:

   $$\neg ◊\neg p\leftarrow \neg \square ◊\neg p$$

   Then we can use the fact that “it is not possible in any world for not p” is equivalent to “it is necessary that p,”

   $$\square p\leftarrow \neg \square ◊\neg p$$

   Then we use the fact that there is an equivalence between “it is not necessary that p” and “in some possible world not p” to get:

   $$\square p\leftarrow ◊\neg ◊\neg p$$

   Then finally, if it “is not possible that not p” then it must be “necessary that p,” and so we get:

   $$\square p\leftarrow ◊\square p$$

   Just by rearranging, we get our result:

   $$◊\square p\to \square p$$
10. ^{^}

    This is not a claim to originality, I’m sure some other logicians have arrived at the same conclusion — it is not a particularly difficult adjustment to notice.

11. ^{^}

    Although for certain other properties I think you would need to be more careful about the arguments for axioms surrounding necessary existence.