On Tue, Feb 10, 2015 at 12:50 AM, Bruno Marchal <[email protected]> wrote:
> > On 08 Feb 2015, at 05:07, Samiya Illias wrote: > > > > On Thu, Feb 5, 2015 at 8:27 PM, Bruno Marchal <[email protected]> wrote: > >> >> On 04 Feb 2015, at 17:14, Samiya Illias wrote: >> >> >> >> On Wed, Feb 4, 2015 at 5:49 PM, Bruno Marchal <[email protected]> wrote: >> >>> >>> On 04 Feb 2015, at 06:02, Samiya Illias wrote: >>> >>> >>> >>> On 04-Feb-2015, at 12:01 am, Bruno Marchal <[email protected]> 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 [email protected]. >>>> To post to this group, send email to [email protected]. >>>> 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 [email protected]. >>>> To post to this group, send email to [email protected]. >>>> 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 [email protected]. >>>> To post to this group, send email to [email protected]. >>>> 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 [email protected]. >>> To post to this group, send email to [email protected]. >>> 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 [email protected]. >>> To post to this group, send email to [email protected]. >>> 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 [email protected]. >>> To post to this group, send email to [email protected]. >>> 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 [email protected]. >>> To post to this group, send email to [email protected]. >>> 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 [email protected]. >> To post to this group, send email to [email protected]. >> 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 [email protected]. >> To post to this group, send email to [email protected]. >> 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 [email protected]. > To post to this group, send email to [email protected]. > 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 [email protected]. > To post to this group, send email to [email protected]. > 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. 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