On 19 Sep 2012, at 21:51, Stephen P. King wrote:

On 9/19/2012 2:39 PM, Bruno Marchal wrote:
Dear Bruno,

Your remarks raise an interesting question: Could it be that both the object and the means to generate (or perceive) it are of equal importance ontologically?

Yes. It comes from the embedding of the subject in the objects, that any monist theory has to do somehow.

In computer science, the "universal" (in the sense of Turing) association i -> phi_i, transforms N into an applicative algebra. The numbers are both perceivers and perceived according of their place x and y in the relation of phi_x(y).

You can define the applicative operation by x # y = phi_x(y). The combinators are not far away from this, and provides intensional and extensional models.

I remind you that phi_i represent the ith computable function in some effective universal enumeration of the partial computable functions. You can take LISP, or c++ to fix the things.

Bruno
Dear Bruno,

You are highlighting of the key property of a number, that it can both represent itself and some other number.

It is a key property of anything finite, not just number. Lists and strings do this even more easily and naturally.



My question becomes, how does one track the difference between these representations?

By quotations, like when using Gödel number, or quoted list in LISP. Those are computable operations.




You speak of measures, but I have never seen how relative measures are discussed or defined in modal logic.

?

A modal logic of probability is given by the behavior of the "probability one". In Kripke terms, P(x) = 1 in world alpha means that x is realized in all worlds accessible from alpha, and (key point) that we are not in a cul-de-sac world. This gives KD modal logics, with K: = [](p -> q)->([]p -> []q), and D: []p -> <>p. Of course with "[]" for Gödel's beweisbar we don't have that D is a theorem, so we ensure the D property by defining a new box, Bp = []p & <>t.





It seems to me that if we have the possibility of a Godel numbering scheme on the integers, then we lose the ability to define a global index set on subsets of those integers

?


unless we can somehow call upon something that is not a number and thus not directly representable by a number..

?
Not clear. We appeal to something non representable by adding the "& p" in the definition of the modal box, but this is for the qualia and first person notion. The Dt (and variant like DDt, DDBDt, etc.) should give the first person plural, normally. many possibility remains, as the quantum p -> []<>p appears in the three main "material variants" of: S4Grz1, Z1*, and X1*, for p arithmetic sigma_1 proposition (the arithmetical UD).

Bruno

http://iridia.ulb.ac.be/~marchal/



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