HRG

I fully agree with you that in order to establish a mathematical proof one has to go through the whole apparatus of Gödelization and prove more than a few lemmata

However, on the level that philosophical debates are usually held, Quine's version of a Gödel sentence in ordinary language works quite well:

"Yields falsehood when preceded by its own quotation" yields falsehood when preceded by its own quotation

Nice, isn't it ?

Regards,
HRG.

P.S. I've dabbled in "anti-omni" arguments myself, mostly on the lines of Cantor's Power Set Theorem (if S is the set of all statements that God can think about, let's look at the set of all subsets of S ....)

TheJesusConspiracy

HRG,

I'm glad to see someone out there that gets it. Keep up the good work.

--TJC

Jesus Tap-Dancin' Christ

But Y could just as easily be X. The only reason that the statement can never be disproven or proven by X is in virtue that he is X--not any inherrent capabilites of that person, powers, or whatever. A clever trick, but one that proves little in and of itself.

Jesus Tap-Dancin' Christ

Oh, and God CAN prove G. How? In all other manners, he is omnipotent, and therefore may simply temporarily redefine himself in such a manner that for the purposes of the argument alone, he is not God, thereby making him capable of soundly proving G without simultaneously disproving it.

If you allow a being infinite power, you have to account for that within your argument.

Automaton

P is still pure self-reference. P: ~Q v [Q -> Q] would, however, be a valid proposition.

BTW, <a href="http://iidb.org/cgi-bin/ultimatebb.cgi?ubb=get_topic&f=50&t=000355" target="_blank">here</a> you state:

• In order for consciousness to be conscious, it must identify something that exists. Additionally, it must be able to identify something outside of itself. A consciousness that is conscious only of itself is an absurdity, for before it could identify itself as conscious, it would have to be conscious of something.

It is somewhat amusing that you defend one form of pure self-reference here and debunk another elsewhere.

TheJesusConspiracy

Yes, Y could just as easily have been X, which is a point that Godel himself acknowledged.

In any truth system capable of referring to itself, it is possible to construct statements which are true but are not provable in that system. Thus, such systems are incomplete. Examples of such systems are 2nd order predicate logic (logic with quantified predicates and relations), arithmetic, people, computers, and God.

The incompleteness theorem is more than a clever trick. It proves that there are things one system can do that another cannot. Since it applies to God, it is a way of showing necessarily that there are things God cannot do that you and I can.

You're right, though, God can soundly prove the statement J: The Jesus Conspiracy will never prove J. I know that J is true, but I cannot soundly prove it.

To be ridiculously technical, I can prove J, just not soundly prove it.

vide:
1. Jesus was a man and Jesus was not a man.
/Therefore, J.
2. Jesus was a man. (1, simplification)
3. Jesus was a man or J. (2, addition)
4. Jesus was not a man. (1, commutation, simplification)
5. J (3,4, disjunctive syllogism)

This proof of J (provided by me) is valid but not sound.

TheJesusConspiracy

I defend one form because it is provable. The other I debunk because it is not. Self-reference is not a sufficient condition to establish the truth or falsity of a statement.

The statement P: P or ~P is a tautology. It matters not that the only content of P is self-reference. P is necessarily true.

I'm sorry, but you aren't going to be able to win this argument. Your previous example involves a fallacy. You state that God can prove G by redefining himself such that he is not God, and then he can prove G. Well, if he is not longer God, then the former-god-thing can now prove G, but he isn't God at the time of the proving. So your argument fails. Your argument was intended to show that God can do something, but it requires that God not be God in order to do it. Additionally, if you mean that he is God and not-God at the same time, you argument is technically valid (as it has inconsistent premises and is thus an argument from absurdity), but not sound.

Finally, you should note that it still shows that God cannot do something. God cannot prove G without redefining himself and making himself not G, whereas I can prove G without doing so.

Your arguments are completely absurd. I'm not insulting you, though. Absurdity, in logic, means contradiction.

[ June 07, 2002: Message edited by: TheJesusConspiracy ]</p>

TheJesusConspiracy

I don't allow God literally infinite power. Very few people who've thought about it do. I hypothetically assume it for the sake of showing that it is impossible, for literally infinite power entails that power to make itself powerless.

Additionally, the interesting thing about logic is that is self-grounding. If one tries to deny the axioms of logic (identity, contradiction, excluded middle), the meanings of the particles (not, and, or, if-then, if and only if), and the possibility of truth and falsity, one relies upon all of them being unconditionally true, meaningful, and possible, respectively, in order to get one's feet off the ground. Arguments against logic need logic.

Automaton

Huh?
Self-referential statements are not truth claims. There is no way to establish the truth or falseness of them.
It is not necessarily true that it can be the case for P to be ~P. P: P is not a proposition (even if it happens to be the law of identity.) The definition of the proposition is the definition of the proposition. This is the same as having no definition for the proposition. A definition must be given in order for there to be a definition in the first place for it to refer to!

NialScorva

Couple points:

Incompleteness is almost a side effect of what Godel's theorems prove, that the theorems of a formal axiomatic system with finite axioms are both unenumerable and uncomputable except by brute force proofs.

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.

The incompleteness theorems are rather tricky, and do *not* say what many people think they do. It relies on some clever tricks and *many* assumptions about the systems they operate in.

Generally, theists form their broken mathematical arguments using an incomplete understanding modal logic. Atheists generally form their borken mathematical argument from Godel's Incompleteness Theorem. They *both* form broken arguments from Baye's Theorem. Ain't math fun?