Dear Günther,
> thanks for the good references, I will integrate them on the resource > page (or on a separate page). > > Some of these books I have already read (Boolos), You mean read with pencil and paper? Machine's theology has no more secret for you? Have you read the Plotinus paper? > others are on my list > (Rogers). This one is transcendental. Even from the pedagogical and philosophical point of view I think. Only one critic: I prefer the Kleene's version of the second recursion theorem, far better suited for abstract biology (like in my planaria paper). But it is a three line reasoning to go from one form to the another. Note that Kleene version is more general: it works on the subcreative sets, meaning that it can have something to say on tractability issues too. > > > Smullyan's Forever Undecided is unfortunately out of print, but I am > on > the lookout for used copies ;-) > > Best Wishes and thanks for your time in thinking about the best > references, > Günther > > P.S.: I agree with you that the best way to convey knowledge is > discussion - I will keep bugging you with questions concerning COMP > and > UDA *grin* Please do. With pleasure, Bruno > > > Bruno Marchal wrote: >> Günther, >> >> >> AUDA is based on the self-reference logic of axiomatizable or >> recursively enumerable theories, of machine. Those machines or >> theories >> must be rich enough. In practice this means their theorems or beliefs >> are close for induction.This is the work of Gödel and followers, >> notably >> Löb, who found a nice generalzation of Gödel's theorem and Solovay >> who >> proves the arithmetical completeness of the logic he will call G >> and G'. >> Here is the key paper: >> >> Solovay, R. M. (1976). Provability Interpretation of Modal >> Logic. /Israel Journal of Mathematics/, 25:287-304. >> >> I follow Boolos 1979 and Smullyan "Forever Undecided" in calling such >> system G and G*. G has got many names K4W, PrL, GL. >> There are four excellent books on this subject: >> >> Boolos, G. (1979). /The unprovability of consistency/. Cambridge >> University Press, London. >> >> This is the oldest book. Probably the best for AUDA. And (very >> "lucky" >> event) it has been reedited in paperback recently; I ordered it, >> and I >> got it today (o frabjous day! Callooh! Callay! :). It contains a >> chapter on the S4 intensional variant of G, and the theorem (in my >> notation) that S4Grz = S4Grz*. The first person is the same from the >> divine (true) view and the terrestrial (provable) view. >> I have it now in three exemplars but two are wandering. >> >> Boolos, G. (1993). /The Logic of Provability/. Cambridge University >> Press, Cambridge. >> >> This is the sequel, with the Russians' solutions to virtually all >> open >> problems in Boolos 1979. The main problem was the question of the >> axiomatizability of the first-order extension of G and G* (which I >> note >> sometimes qG and qG*). And the answers, completely detailed in >> Boolos' >> book, are as negative as they can possibly be. qG is PI_2 complete, >> and >> qG* is PI_1 complete *in* the Arithmetical Truth. The divine >> intelligible of Peano Arithmetic is far more complex than Peano >> Arithmetic's ONE, or God, in the arithmetical interpretation of >> Plotinus. >> >> >> Smoryński, P. (1985). /Self-Reference and Modal Logic/. Springer >> Verlag, >> New York. >> >> I have abandon this one sometimes ago, because of my eyes sight >> defect, >> but with spectacles I have been able to distinguish tobacco product >> from >> indices in formula, and by many tokens, it could be very well >> suited for >> AUDA. The reason is that it develops the theory in term of >> (computable) >> function instead of assertions, showing directly the relation between >> computability and SIGMA_1 provability. Nice intro from Hilbert's >> program >> to Gödel and Löb's theorem, and the Hilbert Bernays versus Löb >> derivability conditions. It contains a chapter, a bit too much >> blazed in >> the tone, on the algebraic approach to self-reference, which indeed >> initiates originally the field in Italy (Roberto Magari). >> It contains also chapter on the Rosser intensional variants. >> >> Smullyan, R. (1987). /Forever Undecided/. Knopf, New York. >> >> This is a recreative introduction to the modal logic G. I was used >> some >> times ago in this list to refer to that book by FU, and I don't >> hesitate >> to use some of Smullyan's trick to ease the way toward self- >> reference. >> It helps some, but can irritate others. >> Note that Smullyan wrote *many* technical books around mathematical >> self-reference, Gödel's theorems in many systems. >> >> Modal logic is not so well known that such book can presuppose it, >> and >> all those books introduce modal logic in a rather gentle way. But all >> those books presuppose some familiarity with logic. Boolos Et Al. >> is OK. >> It is difficult to choose among many good introduction to Logic. By >> some >> aspect Epstein and Carnielly is very good too for our purpose. >> >> Note that the original papers are readable (in this field). All >> this for >> people who does not suffer from math anxiety which reminds me I >> have to >> cure Kim soon or later. The seventh step requires some math. AUDA >> requires to understand that those math are accessible to all >> universal >> machine 'grasping" the induction principle, this is the work of >> Gödel >> and Al. >> >> I think the book by Rogers is also fundamental. Cutland's book is >> nice, >> but it omits the study of the Arithmetical Hierarchy (SIGMA_0, >> PI_0, SIGMA_1, PI_1, SIGMA_2, PI_2, ...). >> >> >> AUDA without math = Plotinus (or Ibn Arabi or any serious and >> rational >> mystic). Roughly speaking. >> >> I will think about a layman explanation of AUDA without math, and >> different from UDA. >> >> Best regards, >> >> Bruno >> >> >> >> >> On 25 Jan 2009, at 18:45, Günther Greindl wrote: >> >>> >>> Hi Bruno, >>> >>>>> Goldblatt, Mathematics of Modality >>>> Note that it is advanced stuff for people familiarized with >>>> mathematical logic (it presupposes Mendelson's book, or Boolos & >>>> Jeffrey). >>>> >>>> Two papers in that book are "part" of AUDA: the UDA explain to the >>>> universal machine, and her opinion on the matter. >>> >>> I would like to add a "guide to AUDA" section on the resources page. >>> Maybe you could specify the core references necessary for >>> understanding >>> the AUDA (if you like and have the time)? >>> >>> Here a first suggestion of what I am thinking of: >>> >>> Boolos Et Al. Computability and Logic. 2002. 4th Edition >>> >>> Chellas. Modal Logic. 1980. >>> >>> Goldblatt, Semantic Analysis of Orthologic and >>> Arithmetical Necessity, Provability and Intuitionistic Logic >>> to be found in Goldblatt, Mathematics of Modality. 1993. >>> >>> >>> >>> What do you think? >>> >>> Best Wishes, >>> Günther >>> >>> >>> >> >> http://iridia.ulb.ac.be/~marchal/ >> >> >> >> >> > > > http://iridia.ulb.ac.be/~marchal/ --~--~---------~--~----~------------~-------~--~----~ You received this message because you are subscribed to the Google Groups "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 -~----------~----~----~----~------~----~------~--~---
