Hi Brian,
On 24 Sep 2012, at 19:44, Brian Tenneson wrote:
Hi Bruno
On Fri, Sep 14, 2012 at 1:20 AM, Bruno Marchal <[email protected]>
wrote:
Hi Brian,
On 13 Sep 2012, at 22:04, Brian Tenneson wrote:
Bruno,
You use B as a predicate symbol for "belief" I think.
I use for the modal unspecified box, in some context (in place of
the more common "[]").
Then I use it mainly for the box corresponding to Gödel's beweisbar
(provability) arithmetical predicate (definable with the symbols E,
A, &, ->, ~, s, 0 and parentheses.
Thanks to the fact that Bp -> p is not a theorem, it can plays the
role of believability for the ideally correct machines.
How come Bp->p is not a theorem?
If it was, Bf -> f would be a theorem, and thus ~Bf would be a
theorem, and this would contradict the second incompleteness theorem.
Or by Löb, if Bf -> f is a theorem, then by necessitation B(Bf->f)
would be a theorem, and by Löb (B(Bp->p)->Bp) and modus ponens, we
would get Bf, and by Bf -> f, we would get f, and the machine would be
inconsistent.
Bf -> f is a theorem of G*, and so its arithmetical interpretation is
true (in the standard model), but this the machine cannot prove.
What are some properties of B and is there a predicate for knowing/
being aware of that might lead to a definition for self-awareness?
Yes, B and its variants:
B_1 p == Bp & p
B_2 p = Bp & Dt
B_3 p = Bp & Dt & t,
and others.
D? B_1? B_2? B_3?
Dp is defined by ~B~p.
B_1 p is defined by Bp & p, etc. (I mean by their arithmetical
interpretation). When I have more time I will give the precise
Solovaytheorem, here or on FOAR.
btw, what is a machine and what types of machines are there?
With comp we bet that we are, at some level, digital machine. The
theory is one studied by logicians (Post, Church, Turing, etc.).
I am also curious as to the definition of a digital machine.
Anything emulable by a Turing machine, or anything definable by a
sigma_1 arithmetical relations. This is provably equivalent, and get
very general with Church thesis.
Is there a generic description for a structure (in the math logic
sense) to have a belief or to be aware; something like
A |= "I am the structure A"
?
Yes, by using the Dx = xx method, you can define a machine having
its integral 3p plan available. But the 1p-self, given by Bp & p,
does not admit any name. It is the difference between "I have two
legs" and "I have a pain in a leg, even if a phantom one". G* proves
them equivalent (for correct machines), but G cannot identify them,
and they obeys different logic (G and S4Grz).
DX = xx?
I will soon explain this with the phi_i notations. But the basic idea
is that you can build a self-referential machine by applying a
duplicator to itself. If Dx produces "xx", DD will produce "DD", that
is itself. It is an effective diagonalization. See how to build
amoeba, and self-regenerating programs with this in my paper "Amoeba,
planaria and dreaming machines", where I illustrate in LISP how to use
that idea. Some subroutine are explained in the appendice of
"conscience & mécanisme".
Finally, on a different note, if there is a structure for which all
structures can be 1-1 injected into it, does that in itself imply a
sort of ultimate structure perhaps what Max Tegmark views as the
level IV multiverse?
A 1-1 map is too cheap for that, and the set structure is a too much
structural flattening. Comp used the simulation, notion, at a non
specifiable level substitution.
This structure I have in mind having the property that all
structures can be injected into it has more structure than a set
structure. See, I have revised my thoughts and put them into a
fairly short document. You helped me a year or two ago to show me
some flaws with my thoughts in a document. I could send it to you.
OK. September is busy, but I will have more time (I hope) in october,
you can send it to me.
Best,
Bruno
http://iridia.ulb.ac.be/~marchal/
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