On 06 Jan 2014, at 22:49, Gabriel Bodeen wrote:
On Monday, January 6, 2014 3:27:49 AM UTC-6, Bruno Marchal wrote:
Hi Gabriel,
On 06 Jan 2014, at 02:48, Gabriel Bodeen wrote:
> Hi Bruno (& all),
> I was trying to read through your paper "The Origin of Physical Laws
> and Sensations", which I saw linked to in a conversation earlier. I
> started to get lost about page 13 of the PDF,
Waw! Good. No problem with the UDA?
Heh, well I can't promise I understood fully and correctly, but that
qualitative section seemed to pass muster in my head. When it gets
down to the math, there's less room for me to be fooling myself.
I am of course interested in knowing whatever you might find unclear
in the UDA, which is supposed to be infinitely simpler than AUDA
(except for professional logicians).
What is it that you don't understand page 13? You might need to study
a good book in logic, like Mendelson, or Boolos and Jeffrey (+
Burgess in late editions).
What is "DU accessibility"?
It means a computational state, in fact. It means all machines' state,
when thought to belong to a computation emulated by the UD. It means a
phi_j(i)^n (the state of the machine j on input i, after n steps of
its computation).
Thanks for the suggested list of books!
The books above can help, but you can also copy and past the first
paragraph that you don't understand, and I can explain more online,
although you might need to make more solid your basic knowledge in
mathematical logic.
Quote:
Going from knowledge to belief makes things much more subtle and
interesting. Indeed the paradox above, for example, will occur only
if the visitor (which the habitant is addressing) believes all his
beliefs are true. In the case where indeed all his beliefs are true,
the reasoning above will show that the reasoner can neither believe,
nor know for the matter, the very fact that all his beliefs are
true. So if all the propositions Bp -> p are true about you, they
cannot all be believed by you. Instead of a paradox, we get an
incompleteness result. And you don’t need really to go on the KK
Island; it is enough some habitant asserts ‘‘Mister X or Misses X
will never believe I am knight.’’ That sentence will be true,
although unbelievable by X, independently of the fact X met such
sentence. Imagine a native saying ‘‘the Belgians will never believe
I am a knight,’’ then any Belgian believing in its own accuracy,
i.e. believing in all the propositions Bp -> p, will be inaccurate,
even if the Belgian didn’t know anything about the KK Island. Giving
that the use of ‘‘believe’’ instead of ‘‘know’’ evacuates the
paradox, such an island could well exist and the assertion of their
inhabitants could have consequences on our ability or inability to
believe some truth! This is a very weird situation. To reassure
ourselves we can still hope such an island does not exist.
I am just saying that the sentence "you will never know that this
sentence is true" is contradictory, when said to a correct reasoner
having enough logical reasoning ability. If the sentence is true, you
will never know that it is true, given that this is what the sentence
says. So if it is false, you will know that it is true, but we suppose
you are correct, so you cannot know that it is true, but then it has
to be false, so it is true and false. the conclusion of this is that
"knowing" (like "correct") cannot be define by the machine. But if you
replace "know" by "believe (or by "prove")": the situation is no more
contradictory: "you will never believe that this sentence is true"
lead only to a truth, which is unbelievable. That is not a
contradiction, but an incompleteness. To get a contradiction, you
would need to know that you are correct (or even just consistent), so
it shows also that consistency and correctness, when true about you,
cannot be believed or prove by you.
OK? Do you see that? "this sentence is unknowable" leads to a
contradiction. "this sentence is unbelievable" leads to incompleteness.
Bruno
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