On 28 Jan 2015, at 18:33, John Clark wrote:
On Wed, Jan 28, 2015 Bruno Marchal <[email protected]> wrote:
> Cantor brought the contradiction by assuming there is a bijection
between N and the set of infinite binary sequences
Yes, and then he showed that such an assumption was incorrect by
producing a infinite binary sequence that did not correspond to any
natural number.
> The procedure that I use does not assume such bijection. On the
contrary, as I said explicitly each finite sequence of digits
generated at any time is admitted as being the initial segment of a
continuum (uncountable sequence).
You can't do hand waving like that in a proof! You've got to show
exactly how that uncountably infinitely long sequence was produced.
> The "01" appearing above is supposed to be an initial segment of
one sequence,
OK, you gave me 2 elements, but what's the third element in this
uncountably infinitely long sequence of yours?
I have never pretended that I can enumerate them. I say only that I
can generate all of them by dovetailing. The easyness comes from the
fact that I generate them all, by zigzagging on their initial
fragment. Each initial segment is admittedly denoting its many
different precisions.
We both agree that the set of binary infinite sequences is
uncountable. Cardinality 2^aleph_0.
When we say that a program generate a real number, it means that it
generates all the initial fragment of that real numbers. Usually, it
means also that it generates only that real number. Here the method
consists in generating all of them, in that sense of generating all
individual sequences pieces by pieces.
You can't enumerate the real numbers, but you can still dovetail on
all of them.
> It plays some role in the UDA too.
Then I'm even more happy that I stopped reading at step 3.
Without succeeding explaining why to anyone.
I recall your main error: stopping considering the first person
experience after the duplication, despite computationalism provided an
excellent approximation good enough for our purpose: the content of
the diary.
Take the case of the infinite self-duplication, as it illustrates a
form of "pure randomness":
I describe the content of the diaries at each step, and write 0 and 1
instead of W and M. Let us say that the subject is duplicated in room
0 and 1, and he can distinguish them.
After the first duplication, we get two different first person
experience described in the two diaries:
0
1
After the second duplication, we get the four different first person
experience in four diaries:
00
01
10
11
After the third duplication, we get the height first person
experiences, described in 8 diaries:
000
001
010
011
100
101
110
111
Simple combinatory argument can explain why, when the iteration grows,
most of the 2^n resulting sequence are not algorithmically
compressible, and so the average on the first person views in that
"simple" iteration of self-duplication, is indistinguishable from
"pure randomness".
Now, in front of a universal dovetailer, or elementary arithmetic, the
matter are no more simple at all.
You say that consciousness is the way data feels when treated, but you
are unclear if they must be treated by this or that universal
machine. There are infinities of universal numbers competing below
our substitution level, so what?
Associating consciousness and data treatment is nice, but done by
which machine(s)?
Are you invoking the physical universe?
How can a universal digital machine distinguish a physical universe
from a non physical universe, like a diophantine emulation of a
physical universe (quantum if needed)?
Universal machine cannot know which machines they are, nor which
universal machine support them, but they can find a "bastard
calculus" (as Plato call it in the Timeaeus, and Plotinus in "the two
matters" Ennead).
The computationalist "bastard calculus" is only the relative
computational state probability calculus, which generalizes Everett on
all computations in the Church-Turing sense.
Physics becomes a self-referential statistics on universal numbers
relations. Advantage, thanks to Gödel-Löb-Solovay, we get the
distinction between what will be justifiable and what will be not
justifiable (which was to be expected for consciousness, truth, etc.).
Bruno
John K Clark
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