Re: COMP, Quantum Logic and Gleason's Theorem

From: Bruno Marchal <>
Date: Wed, 28 Jan 2009 20:23:50 +0100

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 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
> >

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Received on Wed Jan 28 2009 - 14:23:57 PST

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