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