On 10 Jan 2011, at 00:16, Brian Tenneson wrote:

Those are some very interesting results, I must say.

Thanks.

Why the letter choice of B?

`B is for Gödel's arithmetical provability predicate 'Beweisbar'. It is`

`applied to a number which is coding or expressing a statement.`

Does it "secretly" stand for "believes" or something else?

`It is believability, by any (correct) machine believing in the`

`recursive definition of addition and multiplication (and in not too`

`much more), and is such that if she believes soon or later A, and`

`believes soon or later A -> B, then she will believe B, soon or later.`

`Solovay theorem shows that the modal logic G and G* solves the`

`provability issue, of those machines/theories. At the propositional`

`level. G and G* asserts which statements are provable (and true) or`

`true but unprovable by the machine. G is mainly axiomatized by Lôb`

`formula B(Bp -> p) -> Bp which explains my Löbian machine terminology.`

`Löb formula expresses a form of extreme modesty. If you replace p by`

`the constant false, f, you get Gödel incompleteness theorem: ~Bf`

`(consistency) -> ~B(~Bf). The consistency of the machine entails the`

`unprovability of the self-consistency by the machine.`

`I hope you see that B(Bf -> f) -> Bf is equivalent with Dt -> ~BDt, or`

`that Dt -> DBf. D is ~B ~.`

`D is for Diamond (he modal dual of B, for the modal box. The modal box`

`B of G (and G*) captures completely the logic of Beweisbar B. G`

`capture the part provable by the machine. And G* captures the truth,`

`like Dt, DDt, DBf, etc.`

If x is true then ~Bx seems to imply that if a hypothesis is truethen science will never prove it no matter how much time you givethem (barring any sort of infinite time). Is that right?

Science never proves. It asserts hypothesis, like x + 0 = x x + s(y) = s(x + y) and common inference rules. Science proves things in theories. My theory is the two axioms above + x * 0 = 0 x*s(y) = x*y + x

`and the inside epistemology is given by those four same axioms`

`together with the induction principle on expressible formula (leading`

`to being Löbian).`

I've had some exposure to Alan Watts and all I have seen is bothprofound and simple.

Wow. Well said.

DDDDDt is a bit hard for me to understand. Would you elaborate for me?

D is ~B ~

`Dt is for ~Beweisbar('~t'), that is ~Beweisbar('f'), that is the non`

`provability of the (Gödel number representing the) false, that is`

`(self) consistency, as expressed by the machine. It is third person`

`self-reference.`

`DDDDDt is the consistency of the consistency of the consistency of the`

`consistency of the consistency of t. It is far stronger than just`

`consistency. It is a typical truth (provable by G*), but unprovable by`

`the machine. DDt <-> Dt, but the machine cannot see (rationaly`

`believe) that Dt -> DDt.`

`By Gödel's completeness (not: incompleteness!) we have that`

`consistency (Dt) is equivalent with the existence of a model. Dt is`

`"invention of reality" for a machine. The machine bets instinctively`

`"she does not believe in BS (f, the vulgar name of falsities, like 0 =`

`1.)".`

`You get the arithmetical hypostases by defining new "predicate"`

`connecting believability with truth (first person move) and`

`consistency (material move), eventually iterated. The logic of Bp & p,`

`with p arithmetical gives a non arithmetical knowability or first`

`person view of the machine. You bypass "Tarski theorem" (the non`

`definability of arithmetical truth in arithmetic) by "modeling" the`

`non arithmetical "true('p') by using p itself. That's the very`

`essence of the idea of Theaetetus, and somehow a consequence of the`

`dream argument in metaphysics. Tarski's theory of truth can be relied`

`to this. ##Y*%° is true if it happens that ##Y*%°, or if it is the`

`case that ##Y*%°.`

`The machine cannot prove p <-> Bp, but the machine can prove Bp -> p`

`each and at only time she can prove p, (Löb's theorem); and she can`

`prove p -> Bp for all sigma_1 formula. 'Bp' itself is sigma_1 so that`

`she can prove Bp -> BBp, or dually DDp -> Dp. That is what provides`

`the needed introspection to know that she is universal and that she is`

`incomplete and 'uncompletable', as far as consistent, and that she can`

`crash anytime, and she has already many questions. She can even make`

`money (prove new true theorems) with consistent, but false,`

`proposition (Shit (Bf) can happen: G* proves DBf).`

`I identify a theory with a (chatty) machine, or even a set of beliefs`

`(closed for the inference rules). I study what ideally self-`

`referentially correct universal machine can believe about themselves`

`(Bp) and reality (Dt) and other realities (DBf, DDt, etc.).`

`The logics of Bp & Dt, and Bp & Dt & p, the 'material hypostases',`

`give clues on the way universal machines can succeed in dream sharing`

`and building/filtering realities.`

`Good books are by Boolos (1979, 1993) and Smorynski (1985). Smullyan`

`wrote a recreative introduction to the logic G (Forever Undecided,`

`1987), where 'B' is for believe (by rational believers) (and 'know' is`

`Bp & p, informally).`

`In that 'theology', somehow GOD is already overwhelmed by the DIVINE`

`INTELLECT which is completely overwhelmed by the UNIVERSAL SOUL which`

`is completely overwhelmed by the material (intelligible and sensible)`

`realities. This appears in the first order modal extension of G and`

`G*. Arithmetical truth viewed from inside by universal machines is FAR`

`bigger than Arithmetical truth.`

Bruno

Bruno Marchal wrote:On 05 Jan 2011, at 21:45, Brian Tenneson wrote:"The Tao that can be described is not the ultimate Tao"I <3 Lao tseu, and all the taoists. There is a full chapter on Lao-tseu in the long version of my PhD(*). They were aware of the dreamargument. In fact, I call the modal formula:x -> ~B x (or its contrapositive Bx -> ~x) with B intended forthe scientific communication or proof: the Lao-Tseu Watts Valadierprinciple, there.Alan Watts describes indeed something similar in his book "thewisdom of insecurity", and Valadier, a french jesuit, wrote aremarkable book where it shows that making moral is immoral.Gödel's theorem (and Löb, Solovay) provides many solutions, havingan arithmetical content, for such an equation. Indeed all xbelonging to G* \ G obeys to that equation.With x = Dt (= ~Bf = consistency), you get Gödel's secondincompleteness theorem: Dt -> ~BDt. But DDDDDt is also a solution.Most formula beginning by D (= ~B ~) are solutions. Correctmachines cannot prove that they cannot prove something.Tarski's theorem provides even more insightful solutions, which areanalytical, and on which the correct machine can only be mute.It led me also to a very simple theory of intelligence. A machineis intelligent if she is not stupid, and a machine is stupid ifeither she believes that she is intelligent, or she believes she isstupid. Aagain incompleteness provides solution. From that I showedthat intelligence has a positive feedback on competence, but thatcompetence has a negative feedback on intelligence.Interesting. I wonder if it's so. Whether or not the ultimate Taocan be described has been the object of all my research-relatedthinking for a while now. I finally made a breakthrough this yearon the problem. I still have to manipulate what I think on it andmassage the document about it. At least I can say that I'm nottrying to describe the Tao. I'm trying to describe a descriptionof the Tao. The reduced product of all structures is my candidatefor my description for a description of the Tao.Perhaps Lao Tzu already put in his two cents regarding this kindof TOE.In "conscience and mechanism" I argue in detail that most of thewriting of Lao-Tseu, Tchouang-Tseu, and especially (my favorite)Lie-Tseu can be interpreted by the discourse of the self-referentially correct machine. But Plotinus is closer to us. I havestudied classical chinese and modern chinese, for years, todiscuss on Lao-Tseu with scholars. It is difficult.Mechanism makes a bridge between Smullyan's "Tao is silent" andSmullyan's "Forever undecided". I still don't know if Smullyanwould agree on this. Some remark by him makes me think he is notaware of that connection, or that mechanism favors that connection.If you like Lao-Tseu, you might appreciate Smullyan's book "tao issilent".Bruno (*) http://iridia.ulb.ac.be/~marchal/bxlthesis/consciencemecanisme.html--You received this message because you are subscribed to the GoogleGroups "Everything List" group.To post to this group, send email to everything-list@googlegroups.com.To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com.For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.http://iridia.ulb.ac.be/~marchal/--You received this message because you are subscribed to the GoogleGroups "Everything List" group.To post to this group, send email to everything-l...@googlegroups.com.To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com.For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.

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