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. > >> 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! > > 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 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. > > 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)) 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) Applying axiom 3 Step 3: s(s(s(0))) * s(0) = s(s(s(0))) 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. 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 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)} 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. 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. 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