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
>
> 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)))))))))
>
>
>
>
>
>
>
>> 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 ?
>
> Are you convinced that 768953 * 7999580012 = 768953 + (768953
> * 7999580012) ?
>
If (768953 * 799958001*2*) is corrected to (768953 * 799958001*1*) :)
>
>
>
>
>
>
>> 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))))))
>
>
>
>
>
>
>
>
> 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!
>
> 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.
> 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!
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
>>>>
>>>>
>>>>
>>>>
>>>>
>>>> --
>>>> You received this message because you are subscribed to the Google
>>>> Groups "Everything List" group.
>>>> To unsubscribe from this group and stop receiving emails from it, send
>>>> an email to everything-list+unsubscr...@googlegroups.com.
>>>> To post to this group, send email to everything-list@googlegroups.com.
>>>> Visit this group at http://groups.google.com/group/everything-list.
>>>> For more options, visit https://groups.google.com/d/optout.
>>>>
>>>>
>>>> --
>>>> You received this message because you are subscribed to the Google
>>>> Groups "Everything List" group.
>>>> To unsubscribe from this group and stop receiving emails from it, send
>>>> an email to everything-list+unsubscr...@googlegroups.com.
>>>> To post to this group, send email to everything-list@googlegroups.com.
>>>> Visit this group at http://groups.google.com/group/everything-list.
>>>> For more options, visit https://groups.google.com/d/optout.
>>>>
>>>>
>>>> http://iridia.ulb.ac.be/~marchal/
>>>>
>>>>
>>>>
>>>>
>>>> --
>>>> You received this message because you are subscribed to the Google
>>>> Groups "Everything List" group.
>>>> To unsubscribe from this group and stop receiving emails from it, send
>>>> an email to everything-list+unsubscr...@googlegroups.com.
>>>> To post to this group, send email to everything-list@googlegroups.com.
>>>> Visit this group at http://groups.google.com/group/everything-list.
>>>> For more options, visit https://groups.google.com/d/optout.
>>>>
>>>
>>>
>>> --
>>> You received this message because you are subscribed to the Google
>>> Groups "Everything List" group.
>>> To unsubscribe from this group and stop receiving emails from it, send
>>> an email to everything-list+unsubscr...@googlegroups.com.
>>> To post to this group, send email to everything-list@googlegroups.com.
>>> Visit this group at http://groups.google.com/group/everything-list.
>>> For more options, visit https://groups.google.com/d/optout.
>>>
>>>
>>> http://iridia.ulb.ac.be/~marchal/
>>>
>>>
>>>
>>>
>>> --
>>> You received this message because you are subscribed to the Google
>>> Groups "Everything List" group.
>>> To unsubscribe from this group and stop receiving emails from it, send
>>> an email to everything-list+unsubscr...@googlegroups.com.
>>> To post to this group, send email to everything-list@googlegroups.com.
>>> Visit this group at http://groups.google.com/group/everything-list.
>>> For more options, visit https://groups.google.com/d/optout.
>>>
>>>
>>> --
>>> You received this message because you are subscribed to the Google
>>> Groups "Everything List" group.
>>> To unsubscribe from this group and stop receiving emails from it, send
>>> an email to everything-list+unsubscr...@googlegroups.com.
>>> To post to this group, send email to everything-list@googlegroups.com.
>>> Visit this group at http://groups.google.com/group/everything-list.
>>> For more options, visit https://groups.google.com/d/optout.
>>>
>>>
>>> http://iridia.ulb.ac.be/~marchal/
>>>
>>>
>>>
>>>
>>> --
>>> You received this message because you are subscribed to the Google
>>> Groups "Everything List" group.
>>> To unsubscribe from this group and stop receiving emails from it, send
>>> an email to everything-list+unsubscr...@googlegroups.com.
>>> To post to this group, send email to everything-list@googlegroups.com.
>>> Visit this group at http://groups.google.com/group/everything-list.
>>> For more options, visit https://groups.google.com/d/optout.
>>>
>>
>>
>> --
>> You received this message because you are subscribed to the Google Groups
>> "Everything List" group.
>> To unsubscribe from this group and stop receiving emails from it, send an
>> email to everything-list+unsubscr...@googlegroups.com.
>> To post to this group, send email to everything-list@googlegroups.com.
>> Visit this group at http://groups.google.com/group/everything-list.
>> For more options, visit https://groups.google.com/d/optout.
>>
>>
>> http://iridia.ulb.ac.be/~marchal/
>>
>>
>>
>>
>> --
>> You received this message because you are subscribed to the Google Groups
>> "Everything List" group.
>> To unsubscribe from this group and stop receiving emails from it, send an
>> email to everything-list+unsubscr...@googlegroups.com.
>> To post to this group, send email to everything-list@googlegroups.com.
>> Visit this group at http://groups.google.com/group/everything-list.
>> For more options, visit https://groups.google.com/d/optout.
>>
>
>
> --
> You received this message because you are subscribed to the Google Groups
> "Everything List" group.
> To unsubscribe from this group and stop receiving emails from it, send an
> email to everything-list+unsubscr...@googlegroups.com.
> To post to this group, send email to everything-list@googlegroups.com.
> Visit this group at http://groups.google.com/group/everything-list.
> For more options, visit https://groups.google.com/d/optout.
>
>
> http://iridia.ulb.ac.be/~marchal/
>
>
>
> --
> You received this message because you are subscribed to the Google Groups
> "Everything List" group.
> To unsubscribe from this group and stop receiving emails from it, send an
> email to everything-list+unsubscr...@googlegroups.com.
> To post to this group, send email to everything-list@googlegroups.com.
> Visit this group at http://groups.google.com/group/everything-list.
> For more options, visit https://groups.google.com/d/optout.
>
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to everything-list+unsubscr...@googlegroups.com.
To post to this group, send email to everything-list@googlegroups.com.
Visit this group at http://groups.google.com/group/everything-list.
For more options, visit https://groups.google.com/d/optout.
