Gödel’s Mathematical Proof of God's Existence

Kurt Gödel’s laughable mathematical proof of the divine

In March of 1976, my advisor and favorite history professor, Edward Cashin, threw an end-of-quarter party for his students. At the party, one of the students learned I was an atheist and told me he had a mathematical proof of God’s existence.

Okay, what it is? I asked.

Unfortunately, he couldn’t remember it at the moment. Although he had it written down, he didn’t have the paper with him.

I offered that if the argument was to the effect that the existence of mathematics (as something beyond and outside of us) entailed the existence of a mind (God) beyond and outside of us, I was prepared to refute his proof.

But no, he said, it wasn’t that. It was a comprehensive logical-step by logical-step proof that I wouldn’t be able to refute.

Unfortunately, we never ran into each other again.

Logical proofs of God’s existence have been around for a long time, including Avicenna’s 10th century proof, Anselm’s 11th century ontological argument, Descartes 17th century efforts and Gottfried Liebniz a few decades later.

Around 1941 the famous mathematician, Kurt Gödel, began working on an improved version of Leibniz’s proof, though it wouldn’t be revealed to the world for another 20 years. Here I’ll review Gödel’s effort as presented in the October 4, 2022 issue of Scientific American, Can God Be Proved Mathematically? by Manon Bischoff.

There are 12 steps to the argument, so let’s dive in.

He starts with an axiom—an assumption, in other words: If φ has the property P and from φ always follows ψ, then ψ also has the property P. For simplicity, we can assume that P stands for “positive”. For example:, if a fruit is delicious, a positive property, then it is also fun to eat. Therefore, the fun of eating it is also a positive property.¹

Already the argument is in trouble. Blueberries are delicious, therefore they are always fun to eat? Even when unripe or half-rotten?

True enough, blueberries—as I imagine them—are always delicious. But blueberries in the physical world don’t always live up to my mind’s expectations. Blueberries develop on the bush and are only delicious (if they are) for a short time between unripeness and rottenness.

Another example: If living things (φ) have the property of aliveness (P), and if death (ψ) is inevitable for living things (φ) (“from φ always follows ψ”), then death (ψ) has the property of aliveness (P).

Gödel’s mental categories don’t fit the biological world.

Okay, you might think, but God’s not part of the biological world. As long as the axiom is valid for God’s world, it’s okay. Right?

Nope. First, because we don’t know upfront that “God’s world” exists—that’s what we’re trying to prove. And if the proof we’re utilizing to prove God exists isn’t valid for the biological world, which we know does exist, why presume that it’s valid?

Gödel’s first axiom may always be true in the world of logic, but that isn’t the case for the actual physical, ever-changing world. Which is why science has to be empirical rather than rational when it comes to reality. There is a place for rationality and logic: to insure that our models are self-consistent and communicable, but whether the model fits reality is always an empirical question, not a rational one. And if God is part of reality, the same applies.

And this brings up another point: things in the actual physical world don’t have “properties” in themselves. Their properties come from our minds. Blueberries may be delicious to me, but not to my dog, or not to a flounder, no matter how ripe the blueberries be.

Even ripeness is a property bestowed upon fruit by the bodies & brains of specific species.

At this point, we can already see that Gödel’s proof is in serious trouble. But let’s give it a chance.

The second axiom further sets a framework for P. If the opposite of something is positive, then that “something” must be negative. Thus, Gödel has divided a world into black and white: Either something is good or bad. For example, if health is good, then a disease must necessarily be bad.²

For whom? Disease may be bad for me, but is it bad for the coronavirus?

I may find prime rib delicious, but the dead cow I’m eating probably objected to ending up on my plate. What’s good for me is bad for it.

As I said earlier, properties are assigned by brains/minds. Thus properties we think of as good or bad are necessarily experiences conveyed by third parties. Things are good or bad for them, from their perspective. And different species typically have different perspectives (especially when it comes to eating or being eaten).

Gödel assumes that all properties are “first party” properties—that they are innate in things, rather than assigned by the brains/minds of specific species. Water is wet. But wetness is an experience my body/ mind creates for me; it’s not a property of water. Healthy/Unhealthy, Good/Bad, Positive/Negative, these are never properties of things, but third party judgments by various bodies/minds.

Our estimation of Gödel is sinking rapidly by the minute. But let’s give him a chance to pull out of this nose-dive.

With these two premises, Gödel can derive his first theorem: If φ is a positive property, then there is a possibility that an x with property φ exists. That is, it is possible for positive things to exist.³

With flawed premises, you can only derive flawed theorems. Yes, it’s possible for positive things to exist—as a judgment by a body/mind. Gödel’s confusion about where “properties” of this sort exist makes him look silly.

A “positive” judgment about something doesn’t give that something the innate property of being “positive”. Its “positiveness” is a biased third party judgment. Blueberries are delicious! Maggots are delectable!

Now the mathematician turns for the first time to the definition of a divine being: x is divine if it possesses all positive properties φ. The second axiom ensures that a God defined in this way cannot have negative characteristics (otherwise one would create a contradiction).⁴

This explodes on Gödel even before it hits the ground.

Since in reality these so-called “positive properties” are third party judgments, contradictions are unavoidable in our actual reality. What is positive to the carnivore is negative to its prey.

A God who created an ideal world for carnivores will necessarily be seen negatively by those being eaten, no matter the level of carnivorian worship. Since the “properties” in question are third party opinions, and since there are multiple third parties, it is not contradictory at all for a divine being to be judged positive and negative, good and bad, at the same time. These are outside judgments, not innate properties.

The third axiom states that divinity is a positive characteristic. This point is not really arguable because divinity combines all positive characteristics.⁵

This is starting to turn into proof that God can’t exist. Ask yourself—What possible being who created our world can always and only be assigned “positive” properties from third parties?

What’s good for the lion is bad for the lamb. Any creator of a world where lambs are eaten by lions is inevitably going to be judged less positively by the former than the latter. As a primate, I’m never going to have a high opinion of the creator of the ebola virus. Its creation was a bad idea (although the virus would disagree, if it could).

Gödel argument is laughably bad. And it’s bad from the start. And it’s bad because he failed to understand that properties are assigned by bodies/minds—they are always third-party assignments.

Let’s continue with the presentation of his proof, although it’s already crashed and burned.

The second theorem now becomes a bit more concrete: by combining the third axiom (divinity is positive) and the first theorem (there is the possibility that something positive exists), a being x could exist that is divine.⁶

Divine for the lion perhaps; not so divine for the wildebeest running from the lion.

Gödel’s goal now is to show in the following steps that God must necessarily exist in the framework that has been laid out. For this purpose, he introduces in the second definition the “essence” φ of an object x, a characteristic property that determines all other characteristics. An illustrative example is “puppylike if something has this property, it is necessarily cute, fluffy and clumsy.

“The fourth axiom doesn’t seem too exciting at first. It simply states that if something is positive, then it is always positive—no matter the time, situation or place. Being puppylike and tasting good, for example, are always positive, whether during the day or at night in Heidelberg, Germany, or Buenos Aires.⁷

Okay, so now we’ve got Gödel eating puppies.

Seriously, this has no connection to the biological world. Blueberries are delicious, apparently, even when unripe or rotten. A person, once healthy, is always healthy. Once puppy-like and young always puppy-like and young.

It reminds me of the creationists who place species into fixed categories and insist that apes can only give birth to apes, humans to humans, and therefore evolution is impossible.

Logical containers can be useful for understanding the real world; however, in order for logical containers to be useful, mental flexibility is required.

Since “time, situation and place” (per Gödel) can’t affect how fast I am (if I have the property of being “fast”), it follows that if I’m fast on a racetrack, then I’ll be fast at the bottom of the Marianas Trench.

It’s a method of reasoning which has no connection to physical reality.

If you have a supernatural worldview (mind before matter), maybe your counter is that reasoning doesn’t need to have any connection to physical reality. The divine is not physical, God is not physical. If I deem God to be “good” then God has the property of “good” and it’s not a judgment of mine.

This may be okay if I’m defining God for use within my model.

But does the model fit reality, that is, does it fit the external world?

Can anything (other than abstract categories) really have the innate property of goodness, or have any innate properties at all?

I realize the supernatural goal is to elevate mind or intelligence to divinity and then to Godhead, and claim this God created all. But at some point we have to expel this model out of our mind and into outside reality, else God only exists in our minds.

But once thrust into outside reality, positive properties like goodness become third party judgments, not innate. This is because properties are our mental creations, and we should want God to be more than just one of our mental creations.

We should want God to really exist outside of us.

Gödel can now formulate the third theorem: if a being x is divine, then divinity is its essential property. This makes sense because if something is divine, it possesses all positive characteristics—and thus the properties of x are fixed.⁸

With this step Gödel is making “properties” essential. A tree has the property of “treeness” and now we’re going to call “treeness” essential to being a tree.

Of course this is wrong. A tree doesn’t need “treeness” to be itself because “treeness” is something which exists not in it but in us: we think about trees and create the “tree” category in our minds.

Trees exist whether we think about them and place them into a category and declare “treeness”—or not. They exist whether we or any thinking animal exists—or not.

Things that exist don’t have innate properties and don’t need properties to exist: in short, properties come from minds and exist only in minds.

The next step relates to the existence of a particular being. If somewhere at least one being y possesses the property φ, which is the essential property of x, then x also exists. That is, if anything is puppylike, then puppies must also exist.⁹

If anything is dragon-like, then dragons must exist.

I just drew a dragon on the paper in front of me. My drawing is dragon-like; therefore dragons exist. This may seem slightly unfair, but remember that it is only in a mind that a property such as “puppylike” exists.

Properties are bits of mental currency, useful (hopefully) for thinking about things in the world, but properties exist in minds, not in things in the world. So properties can’t be essential to existence, and only someone with a supernatural worldview would presume otherwise.

Which is what happened here. Gödel has presumed upfront what he is supposed to be proving. If mind existed before matter, then yes, properties can be essential to existence. But if mind came later, then the things that minds think can’t be essential to the existence of puppies or dragons or anything not mental artifact.

According to the fifth axiom, existence is a positive property. I think most people would agree with that.¹⁰

I hate to be ornery, but even here I can’t agree. The existence of an arsenal of nuclear weapons is not a positive thing. From my perspective, the existence of disease-causing viruses is not a good thing. Nor earthquakes, volcanoes, hurricanes. I’m not keen on the existence of mosquitoes either.

Abstracting the word “existence” from things said to exist—it seems to be a semantic exercise at most.

From this one can now conclude that God exists because this being possesses every positive property, and existence is positive.¹¹

Well, there’s the completed train-wreck, laid out for your entertainment.

You might wonder, do people actually take silly arguments like this seriously?

And the answer is, they do.¹²

They even write about it in Scientific American.

As it turns out, Gödel’s logical inferences are all correct—even computers have been able to prove that.¹³

Logically valid, and yet false. Which happens when your premises have no connection to reality. Consider the following valid syllogism:

All mortal beings are human
Rolly-polly bugs are mortal beings
Therefore rolly-polly bugs are human

The first premise is not factual, but the logical inference is correct. Garbage in. Garbage out.

And this is what Gödel has given us.

This does not settle the final question of the existence of one (or more) divine beings. Whether mathematics is really the right way to answer this question is itself questionable—even if thinking about it is quite exciting.¹⁴

I wouldn’t say that thinking about it is exciting. It’s a garbage pile of confusion.

Starting with deifying properties and pretending they have real (as opposed to imaginary) existence.

Of course I’m not saying that the things we imagine don’t exist. They exist in our minds, in our models—which may or may not have any useful connection to reality outside our minds. To determine if there is a useful connection, empiricism is required.

In short, God’s existence is a factual question, and logical argument can’t settle it.

That logical argument can’t settle it might be a clue. That is, it might be useful evidence for us to consider.

If mind came first, why can’t logical proofs like Gödel’s settle the issue? Why should empiricism be necessary to determine facts? Why doesn’t rationalism suffice?

So how should we proceed?

First you have to understand the two competing models: mind first and matter first. You have to unravel what circumstances and predictions would flow from each. Then you have to examine evidence from the world and see which fits better.

Which, by the way, is how scientists work.

1. Manon Bischoff, Can God Be Proved Mathematically?, Scientific American, October 4, 2022. We are told, “This article originally appeared in Spektrum der Wissenschaft and was reproduced with permission.”

I should add that a logical proof of God’s existence is an exercise only someone with a supernatural worldview (mind primary or mind before matter) would embrace. Only rationalists, not empiricists, in other words, since empiricists would understand that God’s existence is a factual question, a question about the nature of external reality—not a question which can be answered a priori. We learn immediately that Gödel’s approach is mind first.

2. Ibid.

3. Ibid.

4. Ibid.

5. Ibid.

6. Ibid.

7. Ibid.

8. Ibid.

9. Ibid.

10. Ibid.

11. Ibid.

12. For example, see https://mindmatters.ai/2021/06/godel-says-god-exists-and-proves-it/ and https://www.templeton.org/news/is-there-a-god-shaped-hole-at-the-heart-of-mathematics?gad_source=5&gad_campaignid=22361053108&gclid=EAIaIQobChMIqMis8duRjQMVPrZaBR3NoSOtEAAYASAAEgIqBvD_BwE and

13. Bischoff, op. cit. See also https://www.fu-berlin.de/en/presse/informationen/fup/2013/fup_13_308/index.html

14. Ibid.