On 20 Dec 2013, at 01:01, LizR wrote:
On 20 December 2013 11:40, meekerdb <meeke...@verizon.net> wrote:
On 12/19/2013 1:30 PM, Jesse Mazer wrote:
To me it seems like "thinking something is true" is much more of a
fuzzy category that "asserting something is true"
Maybe. But note that Bruno's MGA is couched in terms of a dream,
just to avoid any input/output. That seems like a suspicious move
to me; one that may lead intuition astray.
I seem to recall that Bruno claimed this is a "legal" move because
any possible input/output can be encoded as data within the
computation (or something along those lines.
Yes. Eventually it comes to decide what is your "generalized" brain.
If you need the entire physical universe, with 10^100 decimals, that
will change nothing in the reasoning, because in step seven, your
state will still be accessed.
Of course, the entire physical universe also has no input nor output
(by definition of "entire").
For the six first steps, it is easier to assume some high substitution
(neuronal) for the thought experiment. Then in step 7, this "high
level" assumption is eliminated.
No doubt Bruno will be able to explain much better than me).
I have tried to talk in English. Now the fact that we can put the
input in the code is a fundamental theorem for the universal system,
know as the SMN theorems. In terms of the phi_i it means that there is
one function S of two arguments with
phi_i(x) = phi_S(x, 4)() (S10)
or
phi_i (4, y, z) = phi_S(x, 4) (y, z) (S32)
The meta-program "S" take the input (4), and put it in the code, and
suppress one variable.
For example S(4, "READ x, READ y, output x + y") = "Read Y, output 4 +
y".
S is really a substitution.
S is a program, so it exists a number s such that S = phi_s. You can
use this to see that we can write the SMN theorems with quantifying
only on numbers.
The whole of recursion theory can be based axiomatically on the two
axioms:
- SMN "theorem" (here an axiom, "provable" for all reasonable
programming languages, or universal system)
- It exists u such that phi_i(x) = phi_u(i, x) (existence of a
universal number) (again provable for each individual programming
language). The universal function u computes phi_i(x), for any program
i and any data x.
But I guess that here, I do not explain better than you, as I use
"notation", which frighten the beginners or the non mathematicians.
Yet, we need the SMN theorem to explain the Dx = "xx" method (to
define self-reference in arithmetic) in terms of the phi_i and the w_i
(which I promised to do for you!)
But we might need to revise a bit those phi_i and w_i perhaps, but
then I don't want to annoy you with too much technic either. What do
you think? Also we started this on the FOAR list, would you like to
continue this, and on which list? Take it easy. I know we are in an
end of the year feast period :)
Bruno
http://iridia.ulb.ac.be/~marchal/
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