Monday, March 28, 2011

Proving that G-d exists

From Wikipedia, Gödel's proof of G-d's existence:


The proof

Symbolically:

\begin{array}{rl}

\text{Ax. 1.} & \left\{P(\varphi) \wedge \Box \; \forall x[\varphi(x) \to \psi(x)]\right\} \to P(\psi) \\

\text{Ax. 2.} & P(\neg \varphi) \leftrightarrow \neg P(\varphi) \\

\text{Th. 1.} & P(\varphi) \to \Diamond \; \exists x[\varphi(x)] \\

\text{Df. 1.} & G(x) \iff \forall \varphi [P(\varphi) \to \varphi(x)] \\

\text{Ax. 3.} & P(G) \\

\text{Th. 2.} & \Diamond \; \exists x \; G(x) \\

\text{Df. 2.} & \varphi \text{ ess } x \iff \varphi(x) \wedge \forall \psi \left\{\psi(x) \to \Box \; \forall x[\varphi(x) \to \psi(x)]\right\} \\

\text{Ax. 4.} & P(\varphi) \to \Box \; P(\varphi) \\

\text{Th. 3.} & G(x) \to G \text{ ess } x \\
  
\text{Df. 3.} & E(x) \iff \forall \varphi[\varphi \text{ ess } x \to \Box \; \exists x \; \varphi(x)] \\
   
\text{Ax. 5.} & P(E) \\
   
\text{Th. 4.} & \Box \; \exists x \; G(x)
  
\end{array}

Modal logic

The proof uses modal logic, which distinguishes between necessary truths and contingent truths.
truth is necessary if its negation entails a contradiction, such as 2 + 2 = 4; by contrast, a truth is contingent if it just happens to be the case, for instance, "more than half of the earth is covered by water". In the most common interpretation of modal logic, one considers "all possible worlds". If a statement is true in all possible worlds, then it is a necessary truth. If a statement happens to be true in our world, but is false in some other worlds, then it is a contingent truth. A statement that is true in some world (not necessarily our own) is called a possible truth.
property assigns to each object, in every possible world, a truth value (either true or false). Note that not all worlds have the same objects: some objects exist in some worlds and not in others. A property has only to assign truth values to those objects that exist in a particular world. As an example, consider the property
P(s) = s is pink
and consider the object
s = my shirt
In our world, P(s) is true because my shirt happens to be pink; in some other world, P(s) is false, while in still some other world, P(s) wouldn't make sense because my shirt doesn't exist there.
We say that the property P entails the property Q, if any object in any world that has the property P in that world also has the property Q in that same world. For example, the property
P(x) = x is taller than 2 meters
entails the property
Q(x) = x is taller than 1 meter.
The proof can summarized as:
IF it is possible for a rational omniscient being to exist THEN necessarily a rational omniscient being exists.

19 comments:

e said...

good point.

e said...

subscribing

The Real Shliach said...

Um, yeah. Has anyone disproved it?

e said...

I'm laughing out loud (although I'm not quite laughing my ass off).

I was sitting here thinking that I'm going crazy. I don't remember reading this post about god's existence, nor do I remember leaving those comments. I was checking the timestamps on my comments for the tenth time, when I realized the trick. Hehe. CA you are sneaky.

Certified Ashkenazi said...

I actually wasn’t planning to do it originally, but them my wife made me take down the whore post — but then I remembered this.

e said...

What did the Misses dislike about the whore post?

e said...

In the proof, what do the squares and the diamonds symbolize?

The Real Shliach said...

What is this whore post of which you write?

e said...

You snooze you lose.

Certified Ashkenazi said...

What did the Misses dislike about the whore post?

She said that the Rebbe would not want me to discuss whores or — much less — parking wardens. I wrote to the Igros, and the Rebbe told me to listen to my wife. So I did.

TRS: I have no idea. Probably. I think both Descartes and Godel would benefit from learning Chassidus.

e said...

Would the Misses approve of you making fun of the Igros?

As one can learn from Wikipedia, Godel's proof is subject to all the criticisms the traditional ontological proof is subject to. The only ma'aleh of Godel's proof is that he used fancy notation, so that theists can say "Godel proved god's existence mathematically, so it must be valid!"

The Real Shliach said...

Humph. A simple email would suffice.

Certified Ashkenazi said...

A simple e-mail?

The Real Shliach said...

As in, the original post. It sounds interesting.

Mrs. CA said...

It wasn’t interesting. My husband doesn’t like female parking wardens and he expanded upon that dislike.

Certified Ashkenazi said...
This comment has been removed by the author.
The Real Shliach said...

Ahh, all is revealed.

Certified Ashkenazi said...

And now it shall be concealed.

The Real Shliach said...

And there goes that one.