On 10 Feb 2015, at 08:21, Samiya Illias wrote:
On Tue, Feb 10, 2015 at 12:50 AM, Bruno Marchal <marc...@ulb.ac.be>
wrote:
On 08 Feb 2015, at 05:07, Samiya Illias wrote:
On Thu, Feb 5, 2015 at 8:27 PM, Bruno Marchal <marc...@ulb.ac.be>
wrote:
On 04 Feb 2015, at 17:14, Samiya Illias wrote:
On Wed, Feb 4, 2015 at 5:49 PM, Bruno Marchal <marc...@ulb.ac.be>
wrote:
On 04 Feb 2015, at 06:02, Samiya Illias wrote:
On 04-Feb-2015, at 12:01 am, Bruno Marchal <marc...@ulb.ac.be>
wrote:
Then reason shows that arithmetic is already full of life,
indeed full of an infinity of universal machines competing to
provide your infinitely many relatively consistent continuations.
Incompleteness imposes, at least formally, a soul (a first
person), an observer (a first person plural), a "god" (an
independent simple but deep truth) to any machine believing in
the RA axioms together with enough induction axioms. I know you
believe in them.
The lexicon is
p truth God
[]p provable Intelligible (modal logic, G and G*)
[]p & p the soul (modal logic, S4Grz)
[]p & <>t intelligible matter (with p sigma_1) (modal
logic, Z1, Z1*)
[]p & sensible matter (with p sigma_1) (modal logic, X1, X1*)
You need to study some math,
I have been wanting to but it seems such an uphill task. Yet,
its a mountain I would like to climb :)
7 + 0 = 7. You are OK with this? Tell me.
OK
Are you OK with the generalisation? For all numbers n, n + 0 =
n. Right?
Right :)
You suggest I begin with Set Theory?
No need of set theory, as I have never been able to really prefer
one theory or another. It is too much powerful, not fundamental.
At some point naive set theory will be used, but just for making
thing easier: it will never be part of the fundamental assumptions.
I use only elementary arithmetic, so you need only to understand
the following statements (and some other later):
Please see if my assumptions/interpretations below are correct:
x + 0 = x
if x=1, then
1+0=1
x + successor(y) = successor(x + y)
1 + 2 = (1+2) = 3
I agree, but you don't show the use of the axiom: x + successor(y)
= successor(x + y), or x +s(y) = s(x + y).
I didn't use the axioms. I just substituted the axioms variables
with the natural numbers.
And use your common intuition. Good.
The idea now will be to see if the axioms given capture that
intuition, fully, or in part.
Are you OK? To avoid notational difficulties, I represent the
numbers by their degree of parenthood (so to speak) with 0.
Abbreviating s for successor:
0, s(0), s(s(0)), s(s(s(0))), ...
If the sequence represents 0, 1, 2, 3, ...
We can use 0, 1, 2, 3, ... as abbreviation for 0, s(0), s(s(0)),
s(s(s(0))), ...
Can you derive that s(s(0)) + s(0) = s(s(s(0))) with the
statements just above?
then 2 + 1 = 3
Hmm... s(s(0)) + s(0) = s(s(s(0))) is another writing for 2 + 1 =
3, but it is not clear if you proved it using the two axioms:
1) x + 0 = x
2) x + s(y)) = s(x + y)
Let me show you:
We must compute:
s(s(0)) + s(0)
The axiom "2)" says that x + s(y) = s(x + y), for all x and y.
We see that s(s(0)) + s(0) matches x + s(y), with x = s(s(0)), and
y = 0. OK?
So we can apply the axiom 2, and we get, by replacing x (=
s(s(0))) and y (= 0) in the axiom "2)". This gives
s(s(0)) + s(0) = s( s(s(0)) + 0 ) OK? (this is a simple
substitution, suggested by the axiom 2)
But then by axiom 1, we know that s(s(0)) + 0 = s(s(0)), so the
right side becomes s( s(s(0)) +0 ) = s( s(s(0)) )
So we have proved s(s(0)) + s(0) = s(s(s(0)))
OK?
Yes, thanks!
You are welcome.
Can you guess how many times you need to use the axiom "2)" in case
I would ask you to prove 1 + 8 = 9. You might do it for training
purpose.
1+8=9
Translating in successor terms:
s(0) + s(s(s(s(s(s(s(s(0)))))))) = s(s(s(s(s(s(s(s(s(0)))))))))
Applying Axiom 2 by substituting x=8 or s(s(s(s(s(s(s(s(0)))))))),
and y=0,
s(s(s(s(s(s(s(s(0)))))))) + s(0) = s( s(s(s(s(s(s(s(s(0)))))))) + 0)
Applying axiom 1 to the right side:
s(0) + s(s(s(s(s(s(s(s(0)))))))) = s(s(s(s(s(s(s(s(s(0)))))))))
1+8=9
Is the above the correct method to arrive at the proof? I only used
axiom 2 once. Am I missing some basic point?
Let me see. Axiom 2 says: x + s(y)) = s(x + y). Well, if x = 8,
and y = 0, we get 8 + 1, and your computation/proofs is correct, in
that case.
So you would have been correct if I was asking you to prove/compute
that 8 + 1 = 9.
Unfortunately I asked to prove/compute that 1 + 8 = 9.
I think that you have (consciously?) use the fact that 1 + 8 = 8 +
1, which speeds the computation.
Well, later I ill show you that the idea that for all x and y x + y
= y + x, is NOT provable with the axioms given (despite that theorey
will be shown to be already Turing Universal.
No worry. Your move was clever, but you need to put yourself in the
mind of a very "stupid machine" which understand only the axioms
given.
I understand
OK.
Can you show that 1 + 8 = 9. Better, tell me how many times you
will need to use the second axioms?
Nine times. Here:
1+8=9
Prove: s(0)+s(s(s(s(s(s(s(s(0))))))))= s(s(s(s(s(s(s(s(s(0)))))))))
For x=s(0)
Using axiom 2,
Rewriting for y=(s(s(s(s(s(s(s(0)))))))=7
Step 1: s(0)+s(s(s(s(s(s(s(s(0)))))))) =
s{s(0)+s(s(s(s(s(s(s(0)))))))}
Simplifying the bracket on the right side, for
y=(s(s(s(s(s(s(0))))))=6
Step 2: s(0)+s(s(s(s(s(s(s(s(0)))))))) =
s[s{s(0)+s(s(s(s(s(s(0))))))}]
Simplifying the bracket on the right side, for y=(s(s(s(s(s(0)))))=5
Step 3: s(0)+s(s(s(s(s(s(s(s(0)))))))) =
s[s[s{s(0)+s(s(s(s(s(0)))))}]]
Simplifying the bracket on the right side, for y=(s(s(s(s(0))))=4
Step 4: s(0)+s(s(s(s(s(s(s(s(0)))))))) =
s[s[s[s{s(0)+s(s(s(s(0))))}]]]
Simplifying the bracket on the right side, for y=(s(s(s(0)))=3
Step 5: s(0)+s(s(s(s(s(s(s(s(0)))))))) =
s[s[s[s[s{s(0)+s(s(s(0)))}]]]]
Simplifying the bracket on the right side, for y=s(s(0))=2
Step 6: s(0)+s(s(s(s(s(s(s(s(0)))))))) =
s[s[s[s[s[s{s(0)+s(s(0))}]]]]]
Simplifying the bracket on the right side, for y=s(0)=1
Step 7: s(0)+s(s(s(s(s(s(s(s(0)))))))) =
s[s[s[s[s[s[s{s(0)+s(0)}]]]]]]
Simplifying the bracket on the right side, for y=0
Step 8: s(0)+s(s(s(s(s(s(s(s(0)))))))) =
s[s[s[s[s[s[s[s{s(0)+0}]]]]]]]
Using axiom 1
Step 9: s(0)+s(s(s(s(s(s(s(s(0)))))))) = s[s[s[s[s[s[s[s{s(0)}]]]]]]]
Rewriting with round brackets
Step 10: s(0)+s(s(s(s(s(s(s(s(0)))))))) = s(s(s(s(s(s(s(s(s(0)))))))))
OK. (get the feeling you use axiom 2 only 8 times, but that is a
detail).
Let me ask you this. Are you OK with the two following
multiplicative axioms:
3) x * 0 = 0
4) x * s(y) = x + (x * y)
Yes, they hold true when substituted with natural numbers.
Really?
Have you verified for all numbers?
Generalisation ?
Well, I explain to you the type of axioms we need to be able to prove
such generalization. P(n) means P is some formula of arithmetic (made
using only the logical symbols and the arithmetical symbols: they are
s, 0, + and * (together with "(", ")", and as I said the logical
symbol: we can use only "->" (and define ~A by A -> (0 = 1)).
To prove "for all n P(n)", you will need to prove
P(0)
for all n P(n) -> P(n + 1) (P(s(n))
But to extract from this that for all n P(n), we need to explicitly
accept the infinity of induction axioms, for all formula F that we can
express in the arithmetical language.
{F(0) & (for all n F(n) -> F(n + 1)} -> For all n F(n)
Are you convinced that 768953 * 7999580012 = 768953 + (768953 *
7999580012) ?
If (768953 * 7999580012) is corrected to (768953 * 7999580011) :)
Excellent :)
Can you prove that s(s(s(0))) * s(s(0)) = s(s(s(s(s(s(0)))))) ?
This is of course much longer, and you need all axioms 1), 2), 3)
and 4).
I've tried two approaches, but I am getting stuck at the last step.
Please see:
Approach 1:
Prove s(s(s(0))) * s(s(0)) = s(s(s(s(s(s(0))))))
for x=s(s(s(0))) and y=s(0)
Applying axiom 4
Step 1: s(s(s(0))) * s(s(0)) = s(s(s(0))) + (s(s(s(0))) * s(0))
OK. With the usual notation, you proved that 3 * 2 = 3 + (3 * 1)
Simplifying the bracket on the right side, again using axiom 4,
assuming x=s(s(s(0))) and y=0
x * s(y)= x + (x*y)
Step 2: s(s(s(0))) * s(0) = s(s(s(0))) + (s(s(s(0))) * 0)
3 * 1 = 3 + (3 * 0)
Applying axiom 3
Step 3: s(s(s(0))) * s(0) = s(s(s(0)))
3 * 1 = 3
Replacing the value in Step 1:
s(s(s(0))) * s(s(0)) = s(s(s(0))) + s(s(s(0)))
In number terms, this translates to 3 * 2 = 3 + 3 which is correct
but I do not know how to proceed with the proof.
You are just forgetting the axioms 1 and 2. s(s(s(0))) +
s(s(s(0))) matches axiom 2: x + s(y) = s(x + y).
OK?
Step 4: s(s(s(0))) * s(s(0))= s(s(s(0))) + s(s(s(0)))
Using axiom 2,
Simplifying the the right side of the equation, for y=s(s(0))=2
Step 5: s(s(s(0))) * s(s(0)) =s[s(s(s(0))) + s(s(0))]
Simplifying the the right side of the equation, for y=s(0)=1
Step 6: s(s(s(0))) * s(s(0)) =s[s[s(s(s(0))) + s(0)]]
Simplifying the the right side of the equation, for y=0
Step 7: s(s(s(0))) * s(s(0)) =s[s[s[s(s(s(0))) + 0]]]
Using axiom 1,
Step 8: s(s(s(0))) * s(s(0)) =s[s[s[s(s(s(0)))]]]
Rewriting with round brackets
Step 9: s(s(s(0))) * s(s(0)) =s(s(s(s(s(s(0))))))
OK.
Approach 2:
Prove s(s(s(0))) * s(s(0)) = s(s(s(s(s(s(0))))))
for x=s(s(s(0))) and y=0
y = 0 ?
Using the distributive property of multiplication (or whatever is
the correct term for the following),
Step 1: s(s(s(0))) * s(s(0)) = {s(s(s(0))) * 0} + {s(s(s(0))) *
s(0)} + {s(s(s(0))) * s(0)}
This is a bit weird, and besides, I will show that the distributive
law is also NOT provable in this theory.
To prove such generalization, we will later need a quite powerful
collection of axioms: the induction axioms. They made the difference
between "just Turing universal", and "Löbian".
Using axiom 3 to simplify the first {} on the right side,
Step 2: s(s(s(0))) * s(s(0)) = {0} + {s(s(s(0))) * s(0)} +
{s(s(s(0))) * s(0)}
Using axiom 4 to simplify the second and third {} on the right side,
Step 3: s(s(s(0))) * s(s(0)) = {0} + {s(s(s(0))) + [s(s(s(0))) *
0]} + {s(s(s(0))) + [s(s(s(0))) * 0]}
Using axiom 3 to simplify the second and third {} on the right side,
Step 4: s(s(s(0))) * s(s(0)) = {0} + {s(s(s(0))) + 0} + {s(s(s(0)))
+ 0}
Using axiom 1 to simplify the second and third {} on the right side,
Step 5: s(s(s(0))) * s(s(0)) = {0} + {s(s(s(0))) + {s(s(s(0)))}
Removing {},
Step 6: s(s(s(0))) * s(s(0)) = s(s(s(0))) + s(s(s(0)))
which again translates to 3 * 2 = 3 + 3 which is correct but I do
not know how to proceed with the proof.
See above.
Tell me if you are OK with my remarks, and we will proceed (as long
as it is OK with you, of course).
Very OK! Yes, please and thank you!
All right.
The problem with computer science and mathematical logic is that
there is a long and tedious part to do at the beginning, almost like
making empty a sea with a tea spoon.
I've studied a bit of programming/scripting several years ago, so
yes I understand the need to be explicit with all steps.
OK. We have the same need when we translate statement about machine in
arithmetic. That this could be done was illustrated by Gödel, and can
be used to distinguish what is true about a machine (or even more
general creatures) and what the machine (or general creature) can
justify, guess, observe, feel ... about that, admitting some
definition of those terms.
Have you followed my posts on Cantor diagonalization, and Kleene
diagonalization? Do you know what the phi_i are? And the W_i?
I might need also you opinion about how much you conceive that a brain
might be a "natural" machine: which means something between will you
accept digital teleportation in case of some urgent travel?
The question is not a question "in practice", but in theory. Later on
we can prove that no correct machine can determined in a 100%
justifiable way its substitution level, that's why it asks for some
explicit act of faith (which really means: *cannot* be imposed to
someone).
In particular, there are still little other "obvious" idea that you
have used, and which have not made explicit into axioms, so we will
need some refinement.
Please guide. Thanks!
May be after.
The way I proceed makes it possible to avoid wishful thinking, and,
although I find what is there absolutely fabulous, I have learned that
somehow many are not so much interested in that kind of "possible
truth". This seems confirmed by the fact that few like taking salvia
twice, and somehow, I can understand. It is really stuff for those
having a passion in theology.
Have you an opinion on Everett's formulation of quantum mechanics?
Are you OK with meeting another Samiya Illias?
I am not pretending that anything I say is true, but I do say that it
can be derived from computationalism (there is a level of brain/body
description where consciousness is invariant for the functional
digital substitution), and even constructively for "classical
computationalism" (comp + Theaetetus, from the point of view of PA).
Jews, christians, and muslims were more open to, if not more aware of,
the Platonist way to conceive/search God in their earlier periods, but
eventually the mainstream continues to stick on the Aristotelian
dogma, like notably, the existence of some primitively physical beings.
To progress in the fundamental, we have to progress in doubting
*almost* everything. OK?
Bruno
Samiya
In case of doubt, please feel free to ask what is the point of all
this.
Bruno
Samiya
If you can do this, Allah already knows that you are Turing
universal (in some large sense). You can know that too, once we
have a definition of Turing universal.
With computationalism, except for some purely logical axioms, we
have already the "theory of everything". You can see that it has
very few assumptions. It does not assume matter or god, nor
consciousness. The link with consciousness, and Allah, can be made
at some metalevel, by accepting the idea that the brain or the body
is Turing emulable. But for this we need to work a little bit more.
Bruno
Samiya
Bruno
Samiya
to see that this give eight quite different view the universal
machines develop on themselves.
Reminds me of this verse [http://quran.com/69/17 ]:
And the angels are at its edges. And there will bear the Throne
of your Lord above them, that Day, eight [of them].
It is like that: The four first (plotinian) hypostases live
harmonically in the arithmetical heaven:
God
Terrestrial Intelligible Divine
Intelligible
Universal Soul
But then the Universal Soul falls, and you get the (four)
matters, and the "bastard calculus":
Intelligible terrestrial matter Intelligible
Divine matter
Sensible terrestrial matter Sensible Divine
matter
Here divine means mainly what is true about the machine/number
and not justifiable by the numbers.
It provides a universal person, with a soul, consistent
extensions, beliefs, and some proximity (or not) to God (which
is the "ultimate" semantic that the machine cannot entirely
figure out by herself (hence the faith).
Interesting!
All universal machine looking inward discover an inexhaustible
reality, with absolute and relative aspects.
Babbage discovered the universal machine, (and understood its
universality). The universal machine, the mathematical concept,
will be (re)discovered and made more precise by a bunch of
mathematical logicians, like Turing, Post, Church, Kleene.
You are using such a universal system right now, even plausibly
two of them: your brain and your computer. They are a key
concept in computer science. They suffer a big prize for their
universality, as it makes them possible to crash, be lied, be
lost, be deluded. They can know that they are universal, and so
they can know the consequences.
The religion which recognizes the universal machine and her
classical theology might be the one which will spread easily in
the galaxy in the forthcoming millenaries. (Independently of
being true or false, actually).
Bruno
Samiya
If you want to convince me, you have to first convince the
universal person associated to the Löbian machine, I'm afraid.
I am not pretending that the machine theology applies to us,
but it is a good etalon to compare the theologies/religions/
reality-conceptions. The problem is that we have to backtrack
to Plato, where what we see is only the border of something,
that we can't see, but yet can intuit and talk about (a bit
like mathematics or music)
Bruno
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