On 26 Jan 2011, at 11:31, Russell Standish wrote:
On Mon, Jan 24, 2011 at 09:31:23PM +0100, Bruno Marchal wrote:
My point is only that IF we accept digital mechanism THEN the
*appearance* of movement is an inside, first person, construction,
due to the gap between what a machine (number) can prove and what is
true.
It is interesting you say this. Is your reasoning for this that the
logic of Bp
& p enables Kripke frames, which can be identified with the passage of
time?
The logic of Bp & p, that is S4Grz, and its computationalist variant,
S4Grz1 (Bp & p + p -> Bp), enables Kripke frames, like most so called
normal modal logic systems, which appears, in this S4 case, to be a
valuable temporal modal logic. In fact S4Grz is even more temporal, or
"subjective-time temporal (cf Bergson's duration) because S4Grz = S4 +
Grz, and Grz imposes antisymmetry for the relation of accessibility
among worlds/states(*): times seems to fly irreversibly.
But S4Grz enables also intuitionistic logic, which is often related to
a logic of evolving knowledge, and which made Brouwer linking
consciousness and time.
Boolos and Goldblatt discovered independently the arithmetical self-
referential S4Grz. Roughly speaking G proves Bp & p when S4Grz proves
Bp.
What is remarkable is that S4Grz = S4Grz*. The G* (true) level does
not add anything. This explains the confusion between truth and
provability made by the pure (solipsistic) first person (the first
person forgetting the existence of other persons).
Note that in the material hypostases, the one with "& Dt" (or "& Dp"),
we lost the Kripke accessibility, and get topological neighborhoods
instead, which is coherent with physicalness and the continuum of
consistent computational continuation needed for the emergence of the
physical laws.
Bruno
(*) Grz is the rather awkward B(B(p -> Bp) -> b) -> p, discovered
earlier by Sobocynski. Grzegorczyk rediscovered it in the context of
axiomatizing a modal form of propositional intuitionist logic.
http://iridia.ulb.ac.be/~marchal/
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