On 2/8/2011 8:47 AM, Bruno Marchal wrote:

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On 07 Feb 2011, at 20:52, Andrew Soltau wrote:How do you define the relative point of view?Do you know Gödel's provability predicate? The points of view aredefined by intensional variants of the current provability predicateof the machine with or without some oracle. There are 8 basic pointsof view p (truth), Bp (provability/believability), Bp & p(knowability), Bp & Dp (observability), Bp & Dp & p(sensibility/feelability). Three of them inherits the G/G*splitting, making a total of 8. It is really 4 + 4*infinity, becausethe 'material points of view' (with Dp) admits themselves gradedvariants.I know *about* Gödel's provability predicate!Good.(Is the 'intensional' referred to here the 'attach' you used inanother email?)Not really, although it is related."Intensional" refers to the fact that if you define a provable(x) bybeweisbar(x) and x', where x' denote the proposition which has x asGôdel number, you define a probability "predicate",

You mean "provability predicate" don't you?

which is not definable by the machine, or in arithmetic, yet provesexactly the same proposition of arithmetic than the one provable.Provable(x) and beweisbar(x) are intensional variant of provability.They are extensionnally equivalent, but intensionnally different, abit like different algorithm can have the same behavior.More simple beweisbar(x) & ~beweisbar(~x) is an intensional variantof beweisbar(x).Intensional variant of bewesibar(x) have been introduced by Rosser inhis elimination of Gödel's assumption of omega-completeness in theproof of incompleteness of formal systems.I am still no clearer about how you define the machine, "with orwithout some oracle", and what defines the relative point of view.Oracle have been introduced by Turing for the study of the degree ofunsolvability. It is a package of usually infinite information,typically not computable. The halting oracle provides the haltinginformation, that no computer can generate. The goal consisted inshowing that some problem remains non solvable, and that some functionremains uncomputable, even when powerful oracle are added, and thishas been used to study the degrees of unsolvability of arithmeticaland mathematical problems.The UD generate all the oracles, like it dovetails on all the reals(trivial exercise; yet people are often wrong on this because theyconfuse the impossibility of enumerating the reals, with theimpossibility of generating them). Think about the iteratedself-duplication experiment.Given that you are defining 8 basic points of view in the abstract,applied to " intensional variants of the current provabilitypredicate of the machine with or without some oracle", it sounds abit, well, abstract. Could you be a bit more specific?I try to be more specific in sane04. May be we should start from that.Or search hypostasis or hypostases in the archive, or "guardianangel", etc.Read the book by Smullyan, and Boolos 1979 (simpler than Boolos 1993). Read perhaps the Theaetetus by Plato.In short you can say that I model "belief" or "opinion" by "formalprobability" (Bp).

You mean "formal provability"? Mind your "p"s and "v"s. :-)

I define then knowledge, following Theaetetus by the true opinion (Bp& p),

You've never said what your answer is to Gettier's example. Brent

observation by the consistent opinion (Bp & Dp), and sensibility bythe true consistent opinion (Bp & Dp & p). Incompleteness motivatesthe initial model, even if it leads to a restriction on the ideallycorrect machine. The whole thing provides an arithmeticalinterpretation of Plotinus theory of the one, the intellect and thesoul + his double (intelligible and sensible) matter theory. Thearithmetical matter theory has been compared to the current inferredtheory of matter, and it looks, up to now, that Nature is correct :)(correct with respect to comp and its neoplatonist rendering, for sure). Bruno http://iridia.ulb.ac.be/~marchal/

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