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


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 are defined by intensional variants of the current provability predicate of the machine with or without some oracle. There are 8 basic points of 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, because the 'material points of view' (with Dp) admits themselves graded variants.


I know *about* Gödel's provability predicate!


Good.




(Is the 'intensional' referred to here the 'attach' you used in another email?)

Not really, although it is related.

"Intensional" refers to the fact that if you define a provable(x) by beweisbar(x) and x', where x' denote the proposition which has x as Gôdel number, you define a probability "predicate",

You mean "provability predicate" don't you?

which is not definable by the machine, or in arithmetic, yet proves exactly 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, a bit like different algorithm can have the same behavior. More simple beweisbar(x) & ~beweisbar(~x) is an intensional variant of beweisbar(x).

Intensional variant of bewesibar(x) have been introduced by Rosser in his elimination of Gödel's assumption of omega-completeness in the proof of incompleteness of formal systems.





I am still no clearer about how you define the machine, "with or without some oracle", and what defines the relative point of view.

Oracle have been introduced by Turing for the study of the degree of unsolvability. It is a package of usually infinite information, typically not computable. The halting oracle provides the halting information, that no computer can generate. The goal consisted in showing that some problem remains non solvable, and that some function remains uncomputable, even when powerful oracle are added, and this has been used to study the degrees of unsolvability of arithmetical and 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 they confuse the impossibility of enumerating the reals, with the impossibility of generating them). Think about the iterated self-duplication experiment.




Given that you are defining 8 basic points of view in the abstract, applied to " intensional variants of the current provability predicate of the machine with or without some oracle", it sounds a bit, 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 "guardian angel", 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 "formal probability" (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 by the true consistent opinion (Bp & Dp & p). Incompleteness motivates the initial model, even if it leads to a restriction on the ideally correct machine. The whole thing provides an arithmetical interpretation of Plotinus theory of the one, the intellect and the soul + his double (intelligible and sensible) matter theory. The arithmetical matter theory has been compared to the current inferred theory 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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