Dear Stephen,

Kim Jones' post prompts me to ask whether or not a "self-referentially-correct Loebian machine" involves an infinite regress or a non-well founded structure. Given that it is typical to include the idea of a non-prescripted interview, where the questions can have follow ups based on answers given and thus not prespecified, how does a Loebian machine prevent a pathological regress? Is this where one is really coming up with a fancy secular notion of "omniscience" (infinite computational/simulation power)?

   Any idea?

It depends what do you mean by "Omniscient", a word which has different meaning in Applied Logic in AI, Mathematical Logic, Epistemology, and theology.

If by omniscience (of an entity) you are meaning the knowability of all what is knowable, then there is a form of omniscience, of the kind avoid in Artificial Intelligence. This comes from our interest in provABILITY, and knowABILITY, etc. The logiocal consequence of the entities belief/knowledge are believable (knowable). But this is different from knowing all the truth (by incompleteness). Here G* can, at the propositional level, be considered as omniscient, but then remember that can G is not a reasoner reasoning about itself.

Well-foundation is asked for the provability, and the Lob formula B(Bp->p)->Bp can somehow be interpreted as a form of induction axiom, or as a well fondation axiom for the structure of the proof given by the machine/entity. But this has nothing to do with the belief of the machine proper. The basic belief can be the axiom of Peano Arithmetic, or of some Set Theory with an axiom of foundation, or without or with an axiom of antifoundation, etc.

The self-reference of the lobian machines and the lobian angels does not imply infinite regress. Actually it is the original Godel's point that such self-references can be done in a finite way, with finite means, without leading to paradoxes or contradictions. The trick is based on the diagonalization procedures of applying a copying machine to itself. I try to illustrate this sometimes by a little song/puzzle:

if DA gives AA, and DB gives BB, and DC gives CC, what does give DD ?

Best regards,




----- Original Message ----- From: "Kim Jones" <[EMAIL PROTECTED]>
Cc: "Bruno Marchal" <[EMAIL PROTECTED]>; "Everything-List List" <>
Sent: Friday, February 17, 2006 7:05 PM
Subject: Re: belief, faith, truth

Which is very interesting, isn't it? People do seem want the kind of modelled structure for their existence that theology projects. Even though G means we can never know the truth of it, theology tells us it is nonetheless there.

Has anyone on this list read Neale Donald Walsch's "Conversations with God?" series of books? Bruno may well be interested to read at least Volume 1 if he hasn't yet encountered it. The whole book IS the interview with the self-referentially-correct Loebian machine! I realised this yesterday after re-reading sections of it and comparing them to Bruno's thinking.

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