sodium

Here's a kind of curious argument against God's existence. I'm not sure I buy it myself, but it is kind of interesting. It starts on the assumption that there is an omniscient being we'll call God, and then shows that this leads to a contradiction.

Consider the following statement.

God knows that this statement is false.

Now, what is God's opinion of that statement? If he decides that it is true, then he has obviously made an error, because the statement says that he knows it is false. If he decides that it is false, and he is correct, then he knows it to be false, but then the statement is true, so he is in error. God cannot be in error, because he is omniscient.

So, God can't really know whether the statement is true or false. But since God doesn't know the statement to be false, and this is what the statement claims, we know the statement to be false.

So, we know something God does not. But this contradicts God's omniscience, and shows that assuming an omniscient being leads to a contradiction. So, no such being exists. Well, that was easy

Darth Dane

Sodium, did you read my thread, http://www.iidb.org/vbb/showthread.p...threadid=53853 , about this omniscience and contradiction. It is explained somewhat.




DD - Love Spliff

rainbow walking

Hmmm...Well, I'm not omniscient but I'd say a being who is would respond thusly:

God knows that this statement is false.

This sentence contains one subject and two predicates

The statement itself is the subject.

What god knows about the statement is one predicate.

What the statement says about itself is the other.

The statement declares itself to be false and god would concur that what the statement says about itself is true.

The statement also declares that god knows the statement to be false and again god would concur that what the statement says about what god knows is true.

Therefore it is true that the statement is false and that god knows this.

Darth Dane

Rainbow walking: :notworthy :notworthy :notworthy

So eloquently put




DD - Love Spliff

Scrambles

Yeah, Godel actually gets around the apparent self-referential nature of the paradoxical statement:

"This statement is not provable"

via the isomorphism of meta-mathematical concepts to arithmetical relations. So the mathematical equivalent of "This statement is not provable" is provable in a meta-mathematical sense but not at the mathematical level.

I think there's a problem when a statement references itself.


Scrambles

luvluv

Plus, God is three persons. That would help.

Jobar

luvluv, I *think* you meant that as a joke- anyway, I'm laughing!

diana

Me too. I'm in non sequitor heaven.

d

bd-from-kg

Scrambles:

Your title shows that (like most people) you have little idea what Godel’s Incompleteness Theorem actually says.

What Godel showed is that in any sufficiently “powerful” formal axiomatic system (FAS) with a finitary (technically, a recursively enumerable) set of axioms, there is a formal statement with just one free variable (say P(x) which can be interpreted outside the FAS as meaning that if x is a natural number:

(1) The x’th symbol string (in some arbitrary numbering system) is a well-formed formula (WFF) with just one free variable, x, and
(2) The statement obtained by replacing the free variable x in this statement with the number x cannot be proved in the FAS.

[Note: The key to the proof is Godel’s ingenious demonstration that these statements can be translated into the formal language. That is, he showed how to construct (for any sufficiently powerful finitary FAS) an arithmetical formula WFF(x) such that for any integer x, WFF(x) = 0 if and only if the x’th symbol string (in our numbering system) is a well-formed formula; another formula PF(x, y) such that for any integers x and y, PF(x, y) = 0 if and only if the x’th symbol string is a string of WFF’s which constitutes a valid proof of the y’th symbol string, etc. Thus if S is a well-formed formula which is the N’th symbol string, we have within the FAS the statement “There does not exist an x such that PF(x, N)” – or in English, “There is no string of symbols which constitutes a valid proof of S in this FAS.” or more simply, “S cannot be proved in this FAS”.]

Now obviously P(x) will be false for most x, but the interesting cases are those where x happens to be the number of a statement satisfying (i) and (ii). For example, let E(x) be the statement “x is an even number”, translated into the formal language (it doesn’t matter how). Let’s say that E(x) is statement number N (according to our arbitrary numbering system). Then P(N) is true just in case there is no proof in the formal system that N is an even number.

But now notice that P(x) itself is a symbol string with a number – say M. So our meta-interpretation of P(x) leads naturally to the interpretation of P(M) as saying that there is no proof in the formal system of P(M).

Now if P(M) is false, it follows that there is a proof of P(M) in the FAS, which of course is catastrophic for the formal system: it must be logically inconsistent. But if P(M) is true, it follows only that there is no proof of P(M) in that particular FAS, which does not lead to any sort of contradiction. It means only that the FAS is incomplete – i.e., there are statements which cannot be proved or disproved within it.

The Incompleteness Theorem does not entail that there are statements which are true but cannot be proved. It entails only that for any finitary FAS there are statements that cannot be proved or disproved within that system. It is essential to the proof that the formal systems are finitary in a definite sense – i.e., that there is a finite algorithm which (if left running indefinitely) will eventually generate all of the axioms (and another that will generate all of the theorems) of the formal system. This is obviously not true of all true propositions taken as a whole: there is no finitary axiom system that incorporates all of them. So the theorem does not imply (for example) that there are tautologies (mathematical theorems and the like) that cannot be proved.

As should be clear by now, the GIT is not a statement about “the way things are” in some cosmic sense; it is simply a statement about the limitations of finitary formal axiomatic systems.

Now let’s turn to your statement:

(*) God knows that this statement is false.

It should be clear by now that this has nothing to do with the GIT, which does not deal with self-referencing statements, but with formal axiom systems. Indeed, this statement cannot be rendered in any formal axiom system for the simple reason that it is self-referencing. This in itself makes it very dubious whether it actually means anything (i.e., whether it expresses a proposition). Most logicians are inclined to exclude all self-referencing statements from the class of meaningful statements. Certainly ones like “This statement is false” must be excluded. It cannot have a truth value, since if true it is false and if false it is true. So the only possible conclusion is that it is meaningless. Similarly, “I know that this statement is false” cannot express a proposition. For if it is true it must be false. Thus if it expresses a proposition, I know that it’s false, which means that it’s true. Thus it cannot express a proposition – i.e., it cannot have a truth value - and hence is meaningless.

Note that this has nothing to do with God or omniscience; it follows from the logical form of such statements that they must be meaningless. The reason God doesn’t know that (*) is true or false is that it isn’t true or false, just as the reason I don’t know that “I know that this statement is false” is true or false is that it isn’t true or false.

sodium

But the statement says "God knows that this statement is false". If God knows that that statement is false, then that means its true. You can't get around this fact without radically altering the meaning of the statement. However, I'm going to start saying "God believes" instead of "God knows" so that the statement is clearly only talking about what God believes, and not directly about its own truth value.

Now, Scrambles suggests that the problem is that the statement references itself. But with a little work, you can avoid the self-reference.

I'm going to introduce a new concept called Quineing. When you Quine a statement, you replace every "Q" that stands alone as a word with the original statement itself, in quotes, followed by the phrase "when Quined". So, for example,

Q is my cat
when Quined becomes
"Q is my cat" when Quined is my cat

Quineing is just something you can do with any arbitrary string of text. The text doesn't even have to mean anything.

Now, consider the following statement,

God believes "God believes Q is false" when Quined is false

Looks a little like nonsense, but we can determine the meaning. It opens with a bit of text, and claims that if you take this text, and Quine it, you get a false statement (as opposed to a true statement, or a meaningless statement). So it's a statement about another statement. I'll call the former the meta-statement, and the latter the statement.

But what is the statement? By remembering our definition of Quineing we determine that it is none other than the original statement. If I take

God believes Q is false
and replace the Q with
"God believes Q is false" when Quined
I get
God believes "God believes Q is false" when Quined is false

So, I won't bother to work out the possibilities again, but we're in the same mess as before. And the only way out is to declare that the statement is false, but that God does not believe the statement to be false.