Freethought & Rationalism Archive

06-07-2002, 11:22 PM

Jesus Conspiracy,

Quote:

Take the statement, G: "God will never soundly prove G."

1. Assume not-G.
2. If not-G, then God will at some point soundly prove G.
3. If God soundly proves G, then G.
4. From 1-3, G and not-G.
5. Therefore, by indirect proof 1-4, our assumption is necessarily false, and it is the case that G.

Here is an interesting consideration which just struck me:
By the definition of Omnipotence and omniscience, God can solve the halting problem. Since every algorithm can be seen out to it's infinite extent, he has a general purpose way of saying how long and whether a problem can be solved given a system.

Since the very definition of omnipotents demands a solution to the halting problem, God must somehow be able to violate Godel's theorem. A strange and counterintuitive result!

To put it another way, Godel's theorem shows that to prove a system to be complete or consistent, we require a stronger system. Since the Divine System is of infinite power, Godel's result does not and cannot directly apply to him.

Is there any rational reason to believe that God exists and can do any of this? I have seen none.

However, you should remember that should he exist, the odds of being smarter than God are 3,720 to Zero.

Regards,
Syn

[ June 08, 2002: Message edited by: Synaesthesia ]

06-07-2002, 11:32 PM

Nial,

Quote:

Also note that it only applies to finite sets of axioms as well, if one were to postulate that God is not a set of axioms, or that it's not a *finite* set, then the incompleteness theorem doesn't necessarily apply.

<img src="graemlins/banghead.gif" border="0" alt="[Bang Head]" /> I should have read more carefully, this is exactly what I mean to say. At any rate, this shows quite conclusively that God's incompleteness theorem requires a hidden assumption (finite axioms) which eliminates it as a disproof of God.

seebs · 06-07-2002, 11:54 PM

My theory was always that God can't create surrealist art; anything He creates is necessarily *definitive* of the nature of reality.

Obviously, the platypus is the counterexample.

[ June 08, 2002: Message edited by: seebs ]

bd-from-kg · 06-08-2002, 06:45 AM

TheJesusConspiracy:

A clever argument, but it doesn’t quite work.

First, a minor quibble: the argument as it stands doesn’t even purport to prove what you say it does. It purports to show only that God will not prove G. To get a contradiction you need to show that God cannot prove G. So let’s revise G to read:

G': God cannot soundly prove G'.

Now assume for the sake of argument that, notwithstanding the self-referential nature of G', it expresses a proposition. Then an argument parallel to yours can obviously be constructed proving G', to wit:

1. Assume not-G'.
2. If not-G', then God can soundly prove G'.
3. If God can soundly prove G', then G'.
4. From 1-3, G' and not-G'.
5. Therefore G'.

But there’s a problem here. To show what it is, let’s begin by considering what it means to say “X has soundly proved P”.

Strictly speaking, a formal proof is not something that one “does”; it just is. Thus “X has soundly proved P” cannot mean that X has created a sound proof of P; it has to mean something like “X has written out a sound proof of P, or has pointed to a written copy of such a proof”.

Now let’s consider what it might mean to say “X can soundly prove P”. In view of the above, it must mean something like “X can (in principle) write out a sound proof of P or point to a written copy of such a proof.” Since God (being omnipotent by stipulation) could certainly write out any proof that actually exists, that means that G' is logically equivalent to:

G'': There is no sound proof of G'.

Thus, assuming that G' expresses a proposition, we have:

1. You have soundly proved G'.
2. If you have soundly proved G', then G' is true.
3. If G' is true, then G'' is true.
4. If G'' is true you have not soundly proved G'.

Thus the assumption that G' expresses a proposition leads to a contradiction, and so must be false.

There is no parallel refutation of Godel’s Incompleteness Theorem because it says nothing about anyone “proving” anything; it refers only to the existence of a proof.

By the way, I’m not a theist, but I am a logician.

CardinalMan · 06-08-2002, 07:35 AM

TheJesusConspiracy and HRG:

I'm not talking about the proof (which I don't have any more of a problem with than I do with any philosophical argument). I'm talking about the statement of G, which uses G in its definition. If you allow such self-reference, then with imprecise definitions of provable and definable you end up with absurdities such as:

This sentence is false.

The smallest natural number not definable in English using less than one-hundred words.

I see no difference between these statements (the first of which you said was technically meaningless) and your G. In my opinion, this is where all the philosophical arguments using natural language break down. Without precise definitions, they lead to self-refuting statements. Only when we formally define "proof" and "definable", and show how to achieve self-reference without assuming it, do we get rid of absurditites and actually achieve deep theorems.

Now it is true that you can achieve forms of self-reference rigorously (such as in the proof of Goedel's Theorem or in the proof of the Recursion Theorem). However, if you want to claim that the natural language analogues are meaningful, then I think you have to explain the meaning of the above two sentences which refute themselves. This is the problem with natural language (and the reason why we need formal languages to make sense of self-reference, provability, and definability).

CardinalMan

Clutch · 06-08-2002, 08:11 AM

Quote:

Gosh, yawn, what the hell?

Ah, if only we could all be Philosophy students who have taught logic a billion times!

Again, TJC, it is unclear what you think you have an argument for. You can do something God cannot: prove G. I and others have pointed out that your argument is opaque in some important respects. But grant that it works: So what? It is hardly part of the Christian conception of God that his knowledge consists in the possession of proofs, nor that his power is complete under every description of activities.

Seebs' "surrealist" point is a brilliant demonstration of this. Everything that you argue is encapsulated far more elegantly in that point, so far as I can tell. Though, to be sure, I have only taught logic a half-billion times.

An obvious response to your reasoning is: Of course God can't *prove* G. He can't *prove* anything; that would be contrary to his perfect and complete nature. By the same token, here's something else that you can do, but God can't: *Learn* things. Does that impugn his omnipotence? Not at all, since learning is the correction of a defect, ignorance, in just a more general sense than proving is the correction of the same defect. But God is without defect (working within the myth, here); his knowledge timeless and complete; it is not a matter of the activity of proving nor the acquisition of proofs.

(All of which suggests that a better point of attack on provability and God's knowledge is by questioning the notion of an atemporal cognitive agent. This strikes me as being of dubious coherence. )

Why the heck do I keep finding myself arguing on behalf of the theists? Hmm. Probably the same reason why I'm cheering for the Hurricanes to win the Stanley Cup...

[ June 08, 2002: Message edited by: Clutch ]

CardinalMan · 06-08-2002, 10:41 AM

Let me clarify a little.

In order to maintain that some self-referential statements are meaningful and have determinate truth values, while others do not, you have to give some criterion for distinguishing between the two. It seems to me that the reason you believe that "This statement is false" is technically meaningless is because you end with a contradiction if you assume that it is true, and one if you assume that it is false. Why can't the same argument be applied to your G? Since this supposed god can prove everything, we get a contradiction either way (if god can prove G, a clear contradiction, and if god cannot prove G, then we contradict the fact that god can prove everything). Therefore, in your words, G is "technically meaningless" in just the same way as "This statement is false".

CardinalMan

[ June 08, 2002: Message edited by: CardinalMan ]

seebs · 06-08-2002, 12:15 PM

Quote:

Originally posted by Clutch:
 It is hardly part of the Christian conception of God that his knowledge consists in the possession of proofs, nor that his power is complete under every description of activities.

I think that God is omnipotent, but that not all verb clauses in English denote "actions". Thus, saying that God "can't do X" may not actually be describing a limitation on God's power, but rather, showing that language can be meaningless.

I'm rather fond of the "Entity X cannot self-consistently prove this statement" example; it also serves to argue against people who feel that, since machines can't solve the halting problem, humans are "better". (I don't necessarily reject the conclusion; I just reject the argument.)

TheJesusConspiracy · 06-08-2002, 03:17 PM

Quote:

TheJesusConspiracy:
A clever argument, but it doesn’t quite work.

First, a minor quibble: the argument as it stands doesn’t even purport to prove what you say it does. It purports to show only that God will not prove G. To get a contradiction you need to show that God cannot prove G. So let’s revise G to read:
[/qoute]

I am well aware of what this argument accomplishes. It literally shows that there is no point in time in which God will soundly prove G. I would argue that stating God will never prove G is the same as saying God cannot prove G. For those that would counter with something like, "God is timeless so he can prove G outside of time", it would suffice to say that the proof still shows that I can do something God cannot, namely I can exist prove G in time while God cannot. The proof merely shows that God doesn't prove G at any point. The rest is obvious.

[QOUTE]

But there’s a problem here. To show what it is, let’s begin by considering what it means to say “X has soundly proved P”.

Strictly speaking, a formal proof is not something that one “does”; it just is. Thus “X has soundly proved P” cannot mean that X has created a sound proof of P; it has to mean something like “X has written out a sound proof of P, or has pointed to a written copy of such a proof”.

Now let’s consider what it might mean to say “X can soundly prove P”. In view of the above, it must mean something like “X can (in principle) write out a sound proof of P or point to a written copy of such a proof.” Since God (being omnipotent by stipulation) could certainly write out any proof that actually exists, that means that G' is logically equivalent to:

G'': There is no sound proof of G'.

Two points here.

1. To say that x has soundly proven G is to say that x is capable of providing a sound proof of G. That's blindingly obvious. If I ask you if Euclid was able to soundly prove that there are infinite primes, you, being an English speaker anda logician, wouldn't respond, "Well, strickly speaking, no." To say x proves P is to use the verb differently from saying premises 1,2,3 prove P which is to use the word differently from P is a sound proof. This nitpicking isn't even fruitful.

2. People keep pouncing on this omnipotence thing. "Well, obviously if God can do anything then he can do x." The whole point is that this assumption of omnipotence entails contradiction and is thus impossible. Our Christian friends here are arguing with increasing irrelevance, but you should know better as a logician. One might as well say that the first fundamental theorem of calculus is false because we assume it's contradiction in proving it (i.e. going through the process of proof, in case you're getting confused again). The whole point is that some assumptions lead to absurdity and must be thrown out.

Finally, I don't intend this proof to shatter the illusions of theists. It is merely one more kick at the dead horse which is the anti-concept of 'God'.

AdamWho · 06-08-2002, 06:06 PM

Why work so hard to show that "omni-characteristics" are logically inconsistent; it is child�s play.

Can god create a rock that he cannot lift?

06-07-2002, 11:54 PM	#33
seebs Contributor Join Date: May 2002 Location: Saint Paul, MN Posts: 24,524	My theory was always that God can't create surrealist art; anything He creates is necessarily definitive of the nature of reality. Obviously, the platypus is the counterexample. [ June 08, 2002: Message edited by: seebs ]</p>

06-08-2002, 10:41 AM	#37
CardinalMan Junior Member Join Date: Oct 2001 Location: Chicago Posts: 80	Let me clarify a little. In order to maintain that some self-referential statements are meaningful and have determinate truth values, while others do not, you have to give some criterion for distinguishing between the two. It seems to me that the reason you believe that "This statement is false" is technically meaningless is because you end with a contradiction if you assume that it is true, and one if you assume that it is false. Why can't the same argument be applied to your G? Since this supposed god can prove everything, we get a contradiction either way (if god can prove G, a clear contradiction, and if god cannot prove G, then we contradict the fact that god can prove everything). Therefore, in your words, G is "technically meaningless" in just the same way as "This statement is false". CardinalMan [ June 08, 2002: Message edited by: CardinalMan ]</p>

Thread Tools	Search this Thread
Show Printable Version	Search this Thread: Advanced Search

06-08-2002, 06:45 AM	#34
bd-from-kg Veteran Member Join Date: Jul 2000 Location: King George, VA Posts: 1,400	TheJesusConspiracy: A clever argument, but it doesn’t quite work. First, a minor quibble: the argument as it stands doesn’t even purport to prove what you say it does. It purports to show only that God will not prove G. To get a contradiction you need to show that God cannot prove G. So let’s revise G to read: G': God cannot soundly prove G'. Now assume for the sake of argument that, notwithstanding the self-referential nature of G', it expresses a proposition. Then an argument parallel to yours can obviously be constructed proving G', to wit: 1. Assume not-G'. 2. If not-G', then God can soundly prove G'. 3. If God can soundly prove G', then G'. 4. From 1-3, G' and not-G'. 5. Therefore G'. But there’s a problem here. To show what it is, let’s begin by considering what it means to say “X has soundly proved P”. Strictly speaking, a formal proof is not something that one “does”; it just is. Thus “X has soundly proved P” cannot mean that X has created a sound proof of P; it has to mean something like “X has written out a sound proof of P, or has pointed to a written copy of such a proof”. Now let’s consider what it might mean to say “X can soundly prove P”. In view of the above, it must mean something like “X can (in principle) write out a sound proof of P or point to a written copy of such a proof.” Since God (being omnipotent by stipulation) could certainly write out any proof that actually exists, that means that G' is logically equivalent to: G'': There is no sound proof of G'. Thus, assuming that G' expresses a proposition, we have: 1. You have soundly proved G'. 2. If you have soundly proved G', then G' is true. 3. If G' is true, then G'' is true. 4. If G'' is true you have not soundly proved G'. Thus the assumption that G' expresses a proposition leads to a contradiction, and so must be false. There is no parallel refutation of Godel’s Incompleteness Theorem because it says nothing about anyone “proving” anything; it refers only to the existence of a proof. By the way, I’m not a theist, but I am a logician.

06-08-2002, 07:35 AM	#35
CardinalMan Junior Member Join Date: Oct 2001 Location: Chicago Posts: 80	TheJesusConspiracy and HRG: I'm not talking about the proof (which I don't have any more of a problem with than I do with any philosophical argument). I'm talking about the statement of G, which uses G in its definition. If you allow such self-reference, then with imprecise definitions of provable and definable you end up with absurdities such as: This sentence is false. The smallest natural number not definable in English using less than one-hundred words. I see no difference between these statements (the first of which you said was technically meaningless) and your G. In my opinion, this is where all the philosophical arguments using natural language break down. Without precise definitions, they lead to self-refuting statements. Only when we formally define "proof" and "definable", and show how to achieve self-reference without assuming it, do we get rid of absurditites and actually achieve deep theorems. Now it is true that you can achieve forms of self-reference rigorously (such as in the proof of Goedel's Theorem or in the proof of the Recursion Theorem). However, if you want to claim that the natural language analogues are meaningful, then I think you have to explain the meaning of the above two sentences which refute themselves. This is the problem with natural language (and the reason why we need formal languages to make sense of self-reference, provability, and definability). CardinalMan

06-08-2002, 06:06 PM	#40
AdamWho Contributor Join Date: May 2001 Location: San Jose, CA Posts: 13,389	Why work so hard to show that "omni-characteristics" are logically inconsistent; it is child�s play. Can god create a rock that he cannot lift?

Freethought & Rationalism Archive

The archives are read only.