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        general >> truth >> Of Godel and god and religious beliefs
        
(Message started by: BenVitale on Apr 30th, 2008, 1:28pm)
Title: Of Godel and god and religious beliefs
Post by BenVitale on Apr 30th, 2008, 1:28pm
Since Godel's Incompleteness Theorem is valid, we can then demonstrate that it is impossible for
the Bible to be both true and complete.

Now, we know that Godel's First Incompleteness Theorem applies to any consistent formal system, but the trouble i'm having is whether or not religious texts are formal systems.

Your thoughts, please
Title: Re: Of Godel and god and religious beliefs
Post by ThudanBlunder on Apr 30th, 2008, 2:32pm

on 04/30/08 at 13:28:42, BenVitale wrote:
Now, we know that Godel's First Incompleteness Theorem applies to any consistent formal system, but the trouble i'm having is whether or not religious texts are formal systems.

Don't worry, the Bible saves you the trouble by being Completely Inconsistent.   ;)
See attachment.
Title: Re: Of Godel and god and religious beliefs
Post by towr on Apr 30th, 2008, 2:33pm

on 04/30/08 at 13:28:42, BenVitale wrote:
Since Godel's Incompleteness Theorem is valid, we can then demonstrate that it is impossible for
the Bible to be both true and complete.
Godel's incompleteness theory applies to formal systems that prove basic arithmetical truths. The Bible is neither a formal system, nor does it includes arithmetic.

As an example, propositional logic is a consistent and complete formal system.
Title: Re: Of Godel and god and religious beliefs
Post by BenVitale on Apr 30th, 2008, 2:59pm
Is it not possible to apply Godel's Incompleteness Theorem in theological discussions?

However, the logic used in theological discussions is rarely well defined, so claims that Godel's Incompleteness Theorem demonstrates that it is impossible to prove (or disprove) the existence of God are worthless in isolation.
Title: Re: Of Godel and god and religious beliefs
Post by towr on Apr 30th, 2008, 3:15pm

on 04/30/08 at 14:59:04, BenVitale wrote:
Is it not possible to apply Godel's Incompleteness Theorem in theological discussions?
Not unless they include arithmetic among their theological claims.
I suppose you could ask them whether God can determine of any arithmetical statement whether it's true or false. Confident in God's omniscience they are likely to say He can; but Godel proved that impossible.


Quote:
However, the logic used in theological discussions is rarely well defined
It's rarely logic in the formal sense. It's colloquial everyday logic.


Quote:
so claims that Godel's Incompleteness Theorem demonstrates that it is impossible to prove (or disprove) the existence of God are worthless in isolation.
Even if there was a formal theological theory of logic that had something to say about basic arithmetic truth; that still wouldn't imply anything about whether or not the existence of God could be proved in said logic.
The incompleteness of arithmetic doesn't stop someone from proving 1+1=2. Incompleteness doesn't state nothing can be proven in the theory; just that there are statements that can neither be proven nor disproven.
Title: Re: Of Godel and god and religious beliefs
Post by BenVitale on Apr 30th, 2008, 8:28pm
Yes, Godel's First Incompleteness Theorem applies to any consistent formal system which is sufficiently expressive that it can model ordinary arithmetic, and has a decision procedure for determining whether a given string is an axiom within the formal system (i.e. is "recursive").

I'm still trying to figure out whether or not it is possible to build an argument using Godel's theorems to prove or disprove something, take the proposition: god exist. If that proposition is undecidable, the formal system cannot even deduce that it is undecidable. (This is Godel's Second Incompleteness Theorem, which is rather tricky to prove.)

Please bear with me, I need help with this thing.

If you are asking yourselves, what the heck I'm doing with this.

I answer that this is an exercise in reasoning and a chance for me to deepen my understanding of Godel's theorems.
Title: Re: Of Godel and god and religious beliefs
Post by rmsgrey on May 1st, 2008, 4:43am

on 04/30/08 at 15:15:22, towr wrote:
I suppose you could ask them whether God can determine of any arithmetical statement whether it's true or false. Confident in God's omniscience they are likely to say He can; but Godel proved that impossible.

Doesn't Godel's result only apply to the decidability of statements from within the system? A more capable system can prove/disprove the Godel sentence of a less capable system, but will, in turn, have its own Godel sentence. So God may well be able to determine the status of any arithmetical statement - by using something other than arithmetic to do it.
Title: Re: Of Godel and god and religious beliefs
Post by towr on May 1st, 2008, 5:13am

on 05/01/08 at 04:43:43, rmsgrey wrote:
Doesn't Godel's result only apply to the decidability of statements from within the system? A more capable system can prove/disprove the Godel sentence of a less capable system, but will, in turn, have its own Godel sentence. So God may well be able to determine the status of any arithmetical statement - by using something other than arithmetic to do it.
Oh, right; there might be some other (non-arithmetical) undecidable statement instead.
But you could ask God for proofs within the system of arithmetic; of course, at that point it's like asking him to make a square circle. He could tell you it's impossible, but only outside the limits you set.
Title: Re: Of Godel and god and religious beliefs
Post by Grimbal on May 6th, 2008, 6:05am
And God said: "Let 1 plus 1 equal 2".
And so it was.
And God saw that it was good.
:)

Can you prove God does not exist?
The argument could go like this:
God by definition is omnipotent.
If God can do everything, He* can create an enemy powerful enough to defeat Him.  Either He cannot do such a thing, or He can be defeated.  In both cases His power is limited.
So there is no omnipotent God.

It is a like showing that if a system is powerful enough to prove every mathematical truth, it can express a sentence that proves its own inability to prove everything.

*or more likely She.
Title: Re: Of Godel and god and religious beliefs
Post by towr on May 6th, 2008, 6:54am
It depends on what you mean by omnipotent. It seems fair enough to say a being is omnipotent if what (s)he can do is only limited by the logically possible.
I don't see why it's a problem that God might be able to do something that ends his omnipotence though. Until he does he still has every potential.
Title: Re: Of Godel and god and religious beliefs
Post by Mickey1 on Nov 2nd, 2010, 7:27am
A more general question about Godel (non-religious, for which I apologize).

Are we sure the whole width of all theorems can pass his proof, that is, is it not possible that some limitaiton might be inferred from the proof, regarding the nature of  the statements which must be true but but not provable. Could it be e.g. that it refers only to statements about provabilities, for example?

Or can it be proved for example that any statement about provability can be shown to be equivalent to a traditional aritmetic theorem (relation between numbers)?
Title: Re: Of Godel and god and religious beliefs
Post by towr on Nov 2nd, 2010, 7:59am

on 11/02/10 at 07:27:23, Mickey1 wrote:
Or can it be proved for example that any statement about provability can be shown to be equivalent to a traditional aritmetic theorem (relation between numbers)?
Yes, I believe that is exactly what he did; translating an unprovable statement about provability into an arithmetical statement, by using Gödel numbering.
Consequently any logic in which arithmetical statements are possible will contain unprovable statement. (But other logics might not.)


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