So, to take a simple example, let’s say φ is the property “being blue in color”. Axiom 1 says that if being blue in color is a positive property, then being not red is also a positive property, because being blue by definition means you are not red. Axiom 2 then says that being red must not be a positive property, because its negation is positive.
Now we get onto Gödel’s first theorem, which is the following: if φ is a positive property, then it’s possible that something, somewhere, exists that has this property. Seems fair enough.
Next, a juicy definition: Godlikeness. Something is godlike, Gödel says, if it has every possible positive property. What’s more, he says, being godlike is itself a positive property, which we can’t argue with, because he’s made it an axiom.
Gödel now has what he needs to lay out a big theorem: here, he says that since something, somewhere, exists that has every possible positive property, and being godlike is a positive property, that means that something, somewhere, exists which is godlike.
Gödel’s next step is to show that this godlike thing, which we may as well call God, therefore exists everywhere, and he does this by introducing the idea of “essences”.
The first line here defines an “essence” of an object x as a property of x which necessarily implies all other properties of x. So for instance, we might say that “puppyhood” is an essence, because if we know something is a puppy, we automatically also know it’s cute and fluffy and a very good boy or girl.
Gödel then says it’s an axiom that a property being positive somewhere means it’s positive everywhere – arguably true for puppies but perhaps less so for more morally nuanced things like veganism or mimes.
Theorem 3 says that if something is godlike, then that is its defining essence. That pretty much makes sense: being godlike is defined in terms of every possible other property an object can have, so saying something is godlike does indeed tell us everything else we could possibly want to know about it.
He then defines the concept of “necessary existence”. An object exists, he says, if something exists somewhere which has its essential property. So (you’ll be relieved to know) we can logically say that puppies exist, because there are definitely things in the world which have the property of puppyhood (for instance: puppies).
And now for the payoff. Existence, Gödel states, is a positive property. But God has every positive property. And what’s more, something which is positive here is positive everywhere. QED, God exists, says Gödel.
Now, you may have noticed that there are a few problems with this “proof”, and we’ve already touched on the main one: Gödel simply never gave any reasoning for any of his axioms. Mathematically, this means that there are zero reasons to believe his conclusions are true. Philosophically, it means there are zero reasons to believe anything is true. Gödel was a genius, and may have convinced himself of the existence of God, but he certainly didn’t prove it.
Deus Ex Machina
So perhaps a mathematical proof of God is simply too hard for humans to create – but what if we could get a machine to do it for us?
In 2013, two computer scientists made headlines when they uploaded a paper to the preprint server arXiv titled “Formalization, Mechanization and Automation of Gödel's Proof of God's Existence”. They showed – on a MacBook, no less – that Gödel’s conclusion was correct.
At least, assuming his axioms were correct. The truth was that the scientists hadn’t set out to make a theological statement, but a scientific one: all they wanted to do was show off their algorithm.
While many people have attempted over the years to use math to prove the existence of God, nobody has succeeded yet – and it’s unlikely that anybody ever will.