Gödel's ontological argument, step by step

Update 2015: I wrote a new series on ontological arguments.

By request, I am going to explain Gödel's ontological argument step by step, for your reference. Here's how it will work. I am following the symbolic logic proof shown on the Wikipedia article.* In the first section, I will explain everything in English. In the second section, I will write out the proof symbolically, expanding out steps as much as I feel necessary.

Please skip to the section of your preferred level of precision.

*Please note that there are several different variations of Gödel's ontological argument with different sets of axioms and different steps. C. G. Small has another explanation, using a different variation of Gödel's argument.

Section 1: English

All capital letters represent any given property. All small letters represent objects. I use the word "negative" to merely mean non-positive.

• Entailment: V "entails" W if and only if in all possible worlds, all objects with property V also have property W. Note that if in all possible worlds, there is not a single object with property V, then V automatically entails W.
  
  
• Axiom 1: If Z entails Y, and if Z is positive, then Y is also positive.
  
  
• Axiom 2: For all properties Z, either Z is positive or not-Z is positive, but not both.
  
  
• Theorem 1: If a property Q is positive, then in some possible world there exists an object with property Q.
• Proof by contradiction:
• Suppose that Q is positive, and that there is not any possible world where there exists an object with property Q.
• Then in all possible worlds, there are no objects with property Q.
• Therefore Q entails any given property. Q entails R and Q entails not-R.
• Therefore R and not-R are both positive. This contradicts Axiom 2.
  
  
• God-like: An object is "god-like" if and only if the object has every property which is positive. Note that a god-like object cannot have any negative properties. If a god-like object had negative property V, then it would fail to have the positive property not-V.
  
  
• Axiom 3: The property of being god-like is a positive property.
  
  
• Theorem 2: In some possible world, there exists an object which is god-like.
• Since god-like is a positive property, Theorem 1 states that in some possible world there exists an object with the god-like property.
  
  
• Essence: Property V is an "essence" of x if and only if the following conditions hold:
• The object x has the property V.
• If x has any property U, then V entails U.
  
  
• Axiom 4: If a property is positive, then it is positive in all possible worlds.
  
  
• Theorem 3: If an object x is god-like, then the god-like property is the essence of x.
  
• Suppose x has any property Q. x cannot have any negative properties, so Q must be positive.
  
• Therefore, in all possible worlds, Q is a positive property.
• Therefore, in all possible worlds, any god-like object must have property Q.
  
• Therefore the god-like property entails Q.
  
  
• Necessary existence: Object x is "necessarily existing" if and only if the following condition holds: For any property V, if V is an essence of x, then in all possible worlds there exists an object with property V.
  
  
• Axiom 5: Necessary existence is a positive property.
  
  
• Theorem 4: In all possible worlds, there exists a god-like object.
• We know that in some possible world, there exists a god-like object.
• Since necessary existence is a positive property in all possible worlds, that god-like object must be necessarily existing.
• That object has the god-like property as its essence.
  
• By the definition of necessary existence, there must, in all possible worlds, exist an object which has the god-like property.
  
  
• Corrolary 1: If there are two god-like objects, then they cannot have any properties which are different.
• Proof by contradiction:
• Suppose we have two god-like objects, and some property Q which applies to one object, but not the other.
• Since god-like objects cannot have negative properties, Q must be positive.
• Similarly, not-Q must be positive. This contradicts Axiom 2.

Section 2: Symbolic Logic

P represents the property of being "positive". G represents the property of being "god-like". NE represents the property of necessary existence. All other capital letters represent any given property. All small letters represent objects. The letters "ess" represent "is the essence of".

[Update April 2015: I finally converted all these equations to LaTeX. Hope that makes it more readable! Javascript must be enabled.]

• Axiom 1: $\{P(Z) \wedge \square\forall x[Z(x) \Rightarrow Y(x)]\} \Rightarrow P(Y)$
  
  
• Axiom 2: $P(\lnot Z) \Leftrightarrow \lnot P(Z)$
  
  
• Theorem 1: $P(Q) \Rightarrow \Diamond \exists x[Q(x)]$
  
• Proof (by contradiction):
• Suppose $P(Q) \wedge \lnot \Diamond \exists x[Q(x)]$
• $\square\forall x[\lnot Q(x)]$
• $\square\forall x[Q(x) \Rightarrow \lnot R(x)]$
• $P(\lnot R)$ (By Axiom 1)
• $\lnot P(R)$ (By Axiom 2)
  
• $\square\forall x[Q(x) \Rightarrow R(x)]$
• $P(R)$ (By Axiom 1)
• $\lnot P(R) \wedge P(R)$ (Contradiction)
  
  
• Definition of G: $G(x) \Leftrightarrow \forall V[P(V) \Rightarrow V(x)]$
  
  
• Axiom 3: $P(G)$
  
  
• Theorem 2: $\Diamond \exists x[G(x)]$
• Proof: Combine Axiom 3 and Theorem 1
  
  
• Definition of ess: $V~ess~x \Leftrightarrow V(x) \wedge \forall U\{U(x) \Rightarrow \square\forall y[V(y) \Rightarrow U(y)]\}$
  
  
• Axiom 4: $P(Z) \Rightarrow \square P(Z)$
  
  
• Theorem 3: $G(x) \Rightarrow G~ess~x$
• Proof: Suppose $G(x) \wedge Q(x)$
  
• $P(\lnot Q) \Rightarrow \lnot Q(x)$ (By definition of G)
• $Q(x) \Rightarrow \lnot P(\lnot Q)$
• $\lnot P(\lnot Q)$
• $P(Q)$ (By Axiom 2)
• $\square P(Q)$ (By Axiom 4)
• Suppose (for contradiction) that $\lnot \square\forall y[G(y) \Rightarrow Q(y)]$
• $\Diamond \exists y[G(y) \wedge \lnot Q(y)]$
• $\Diamond \exists y[[P(Q) \Rightarrow Q(y)] \wedge \lnot Q(y)]$ (By definition of G)
• $\Diamond \exists y[\lnot P(Q)]$
• $\Diamond \lnot P(Q)$
• $\lnot \square P(Q)$
• $\square P(Q) \wedge \lnot \square P(Q)$ (contradiction)
• $\forall Q\{ Q(x) \Rightarrow \square\forall y[G(y) \Rightarrow Q(y)] \}$
• $G~ess~x$ (By definition of ess)
  
  
• Definition of NE: $NE(x) \Leftrightarrow \forall V[V~ess~x \Rightarrow \square \exists y V(y)]$
  
  
• Axiom 5: $P(NE)$
  
  
• Theorem 4: $\square \exists x G(x)$
• Proof:
• $\Diamond \exists y[G(y)]$ (Theorem 2)
• $\square P(NE)$ (By Axiom 4)
  
• $\Diamond \exists y[G(y) \wedge NE(y)]$ (By definition of G)
• $\Diamond \exists y[G(y) \wedge ( G~ess~y \Rightarrow \square \exists x G(x) )]$ (By definition of NE)
• $\Diamond \exists y[G~ess~y \wedge ( G~ess~y \Rightarrow \square \exists x G(x) )]$ (By Theorem 3)
• $\Diamond \exists y[\square \exists x G(x) ]$
• $\Diamond \square \exists x G(x)$
• Lemma: $\Diamond \square S \Rightarrow \square S$ for any statement S
• $\Diamond \lnot S \Rightarrow \square\Diamond \lnot S$ (an axiom of modal logic)
• $\lnot \square\Diamond \lnot S \Rightarrow \lnot \Diamond \lnot S$
• $\Diamond \square S \Rightarrow \square S$
• $\square \exists x G(x)$
  
  
• Corrolary 1: $G(x) \wedge G(y) \Rightarrow \forall Q[Q(x) \Leftrightarrow Q(y)]$
  
• Proof (by contradiction):
• Suppose $G(x) \wedge G(y) \wedge \exists Q[ Q(x) \wedge \lnot Q(y) ]$
• $P(\lnot Q) \Rightarrow \lnot Q(x)$ (By definition of G)
• $\lnot P(\lnot Q)$
• $P(Q)$ (By Axiom 2)
  
• $P(Q) \Rightarrow Q(y)$ (By definition of G)
• $\lnot P(Q)$
• $\lnot P(Q) \wedge P(Q)$ (Contradiction)

Questions, corrections welcome.

I realize that in the comments, it's hard to use symbols which don't appear on a keyboard. The following symbols will be understood:
~ or $\lnot$ is "not"
[] or $\square$ is "in all possible worlds" or "it is necessary that"
<> or $\Diamond$ is "in some possible world" or "it is possible that"
=> or $\Rightarrow$ is "if ..., then..."
<=> or $\Leftrightarrow$ is "if and only if"
^ or $\wedge$ is "and"
v or $\vee$ is "or"
E or $\exists$ is "there exists"
A or $\forall$ is "for every" or "for all"

Adventures with LaRouchebags

So I have a friend, henceforth referred to as Joe the Skeptic, who likes to talk to the LaRouchebags and other crazy people on campus. It's like a hobby for him.

"LaRouchebags" is our name for the LaRouche movement, a cult of personality around economist and philosopher Lyndon LaRouche. They are a very cryptic and bizarre political movement. If I were to pick an overarching theme, I'd say that it is their opposition to the British Empire. You know, as if we were still fighting the American Revolution.

And because Isaac Newton was British, he couldn't possibly deserve the credit which is given to him. He is a Babylonian witch, and may not have even existed. They believe all of Newton's laws of motion were stolen from Kepler's laws of planetary motion. As a physics student, they would say I'm part of the cult of Newton.

I think these guys prey on college dropouts.

At a previous point in time, Joe the Skeptic had given the LaRouchebags a simple physics problem. The problem involved frictional force. He asked them to solve it through use of Kepler's laws alone, without reference to Newton's laws. Of course, this is impossible, since Kepler's laws only refer to planetary orbits, and say nothing about non-orbital motion. But wouldn't you know, the LaRouchebags never admit it (and never solve the problem either).

The other day, I accompanied Joe the Skeptic on one of his ventures to speak with the LaRouchians. He asked one if he ever solved that physics problem he gave them. Yes, he said, he had e-mailed him the answer. Joe says he never got that e-mail. So the LaRouche guy starts talking about pre-Newtonian concepts of force. Allegedly, such-and-such scientist had decided that F=mv. And then later, another scientist decided that F=mv². So when Newton came up with F=ma, that was nothing new. He just changed a "v" to an "a".

So, I said, Newton changed a wrong equation to a correct one? Doesn't that deserve credit? No, he says. All he did is change a letter. It doesn't matter whether you call it "v" or "a".

So my friends and I started telling him how ridiculous this is. The units aren't correct. "v" and "a" represent completely different concepts. No, they're not measured the same way. Would you say that Einstein simply took the equation E = mv²/2, dropped the 1/2, and changed the "v" to a "c" to make his famous equation E=mc²?

After a bit more of this, the guy Godwinned us all. See, it's people like us that allow this to happen. He points down to the sign on his table, which has a picture of Obama with a Hitler mustache. See, Hitler approves of Obama's health care policies. And if there's one thing we don't like about Hitler, it's his health care policies. Also, all of us are liberals, and liberals can't be scientists!

And then he said he wanted us to go away. He didn't want to talk to us anymore. Joe the Skeptic told me that this was the first time he had seen them ask anyone to go away.

Oh, the stories! Supposedly, many years ago, the LaRouchebags had a formal debate with the campus Objectivist club. Guess who won?