Synaesthesia

Bd-from-kg, Adrian Selby and other's who deny that G is a preposition,

TJC constructed an informal proof (which, given any formal system, could be formalized) that God (defined here as a finite formal system) cannot prove G. He then goes to show that G is true for this finite formal system.

Therefore, G does have a truth value, though not one decidable within the system that G talks about!

In short, there is no problem in TJC's assertion that we can construct a valid preposition like G for every finite formal system.

There is naturally the seperate question of whether God can be characterized by a finite formal system at all. There is the seperate question of whether a formal system god's inability to prove a formally unprovable statement has anything to do with omnipotence.

The truth of G with relation to a formal system cannot therefore constitute a disproof of omnipotence.

CardinalMan

I disagree with this. Most formal systems, for good reason, do not allow direct self-reference. If you allow direct unrestricted self-reference, then you are capable of expressing statements such as "This sentence is false", which would lead to a contradiction in your system (and, if your system allows proof by contradiction, this would allow you to prove everything, which would be pointless).

This statement is true when properly formulated, but TJC's argument is not properly formulated. Since it is a bad idea to allow blatent self-reference, Goedel's Theorem proceeds by simulating self-reference in any first-order formal system with a computable (not necessarily finite) set of axioms which is powerful enough to perform elementary number theory. The argument consists of coding statements and formal proofs as numbers, and using a clever fixed-point lemma to give a statment which indirectly says "This statement is not provable".

I make no claims about whether or not God can be characterized by such a formal system. I am simply trying to point out the unappreciated subtleties involved in Goedel's argument.

CardinalMan

[ June 17, 2002: Message edited by: CardinalMan ]</p>

Adrian Selby

And I don't know enough about formal systems and suchlike to understand what you're talking about

Could you elaborate a touch on what it means to be a finite formal system?

Adrian

Synaesthesia

You're forgetting in your zeal to formalize the godelization of the putative 'Godsystem' that we haven't actually specified what the system is. Therefore to create a godel code is impossible. However, it is reasonable to assume that such a system, like other strong formal systems, is aminable to Godel's procedure.

For this reason, I think it's not only acceptable, it is necessary to use the informal english equivalent. The idea is that we are pointing not to the english sentence G "God cannot prove G", and noting that such a sentence could be constructed within the system.

Although a formalization of this process is impossible here, I appreciate the clarification. That is indeed something I had forgotten.

CardinalMan

Sorry... I'm having trouble getting this to post because of some of the symbols that I've used.

CardinalMan

[ June 17, 2002: Message edited by: CardinalMan ]</p>

CardinalMan

I tried posting a brief description of formal systems here, but had some problems. I've put it up on a web page <a href="http://www.math.uiuc.edu/~mileti/formal.html" target="_blank">here</a>. I've glossed over a lot of details. Please let me know if you have any questions.

CardinalMan