Dear Bruno,
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), others are on my list
(Rogers).
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*
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/
>
>
>
>
>
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Received on Wed Jan 28 2009 - 12:05:10 PST