Le 05-nov.-06, à 00:47, John M a écrit :
>
> Bruno,
> although I did not see in my list-post my comment to Marc's report
> about the
> German conference (sent before your and Saibal's posts) I may continue
> it
> (maybe copying the missing text below);
Your message is in the archive though. See for example:
http://www.mail-archive.com/everything-list.domain.name.hidden/maillist.html
> Saibal's :
> "uncompoutable numbers, non countable sets etc. don't exist in first
> order
> logic,..."
> is interesting: it may mean that the wholeness-view (like Robert
> Rosen's
> 'complexity' and my wholistiv views as well) do not "compute" with 1st
> order
> logic - what may not be fatal IMO.
> I use "model" a bit opposite to Skolem's text (sorry, I could not read
> your
> URL, my new comnputer does not (yet) have ppt installed)
? (Why would you need ppt?)
> - but as I
> decyphered Skolem's long text, it is a math-construct based on 'a
> theory'.
> A different "model".
> My 'model' (and R. Rosen's) is an extract of the totality, a limited
> cut
> from the interconnectedness by topical, functional ideational etc.
> boundaries and THEORIES are based on that (usefully, if not extended
> beyond
> the margins of the model - limitedly observed to formulate them.)
> If this is beyond "1st order logic" that is not the fault of such
> model-view - with uncomputable (impredicative) "numbers" and unlimited
> variables - rather shows a limitation of the domain called 1st order
> logic.
> (I put numbers in quotation, I used the word to apply it according to
> the
> here ongoing talks.)
> Rosen (a mathematician) also called it "Turing un-emulable".
> Your explanation about the ZF uncountability and the uncomputability is
> intgeresting, I could not yet digest its meaning as how it may be
> pertinent
> to my thinking.
To say more on this would need to explain more "logic" (with logic = a
special branch of math).
My point was just that it follows from Church Thesis that the classical
notion of computability is absolute. Like Godel said this is a
mathematical miracle, and my whole work entirely depends on that
miracle (both the "informal but rigorous" UDA, and the formal AUDA).
This is what is lacking in Tegmark for exemple, which take the
mathematical reality for granted (I take only the arithmetical reality
for granted).
But even in the arithmetical frame, the fate of the universal machine
will consist in discovering an absolutely non completely computable
reality, and an infinity (transfinity) of relatively non countable
structures.
The mathematical advantage of comp is that it does not depends of the
notion of order. I interview the PA machine in first order language
because PA speaks fluently in that language, but I could interview
machine talking higher order language as well (even infinitary language
like when I interview "angels" (non turing emulable entity).
I leave the original message below in case you have lost it.
Bruno
>
> John M wrote
>
>
> ----- Original Message -----
> From: "Bruno Marchal" <marchal.domain.name.hidden>
> To: <everything-list.domain.name.hidden>
> Sent: Friday, November 03, 2006 9:08 AM
> Subject: Re: Zuse Symposium: Is the universe a computer? Berlin Nov 6-7
>
>
>
>
> In "conscience et mécanisme" I use Lowenheim Skolem theorem to explain
> why the first person of PA "see" uncountable things despite the fact
> that from the 0 person pov and the 3 person pov there is only countably
> many things (for PA).
> I explain it through a comics. See the drawings the page "deux-272,
> 273, 275" in the volume deux (section: "Des lois mécanistes de
> l'esprit). It explains how a machine can eventually infer the existence
> of other machine/individual). Here:
> http://iridia.ulb.ac.be/~marchal/bxlthesis/Volume2CC/2%20%203.pdf
>
> Note also that the word "model" (in
> http://www.earlham.edu/~peters/courses/logsys/low-skol.htm ) refers to
> a technical notion which is the opposite of a theory. A model is a
> mathematical reality or structure capable of satisfying (making true)
> the theorem of a theory. Like a concrete group (like the real R with
> multiplication) satisfy the formal axioms of some abstract group
> theory. (Physicists uses model and theory interchangeably, and this
> makes sometimes interdisciplinary discussion difficult).
>
> ZF can prove the existence of non countable sets, and still be
> satisfied by a countable model. This means that all sets in the model
> are countable so there is a bijection between each infinite set living
> in the model and the set N of natural numbers. What is happening? just
> that the bijection itself does not live in the model, so that the
> inhabitants of the model cannot "see" the bijection, and this shows
> that uncountability is not absolute. It just means that from where I
> am I cannot enumerate the set. But, contrariwise, "uncomputability" is
> absolute for those "enough rich" theories.
>
> Here I am close to a possible answer of a question by Stathis (why
> comp?), and the answer is that with comp you have robust (absolute,
> independent of machine, language, etc.) notion of everything. Comp has
> a "Church thesis". few notion of math have such facility. Tegmark's
> whole math, for example, is highly ambiguous.
>
> Thanks to Saibal for Peter Suber web page on Skolem (interesting).
>
>
> Bruno
>
>
> Le 03-nov.-06, à 13:50, Bruno Marchal a écrit :
>
>>
>> It is not a question of existence but of definability.
>> For example you can define and prove (by Cantor diagonalization) the
>> existence of uncountable sets in ZF which is a first order theory of
>> sets.
>> Now "uncountability" is not an absolute notion (that is the
>> Lowenheim-Skolem lesson).
>> Careful: uncomputability is absolute.
>>
>> Bruno
>>
>>
>> Le 03-nov.-06, à 13:43, Saibal Mitra a écrit :
>>
>>> uncompoutable numbers, non countable sets etc. don't exist in first
>>> order logic, see here:
>>>
>>> http://www.earlham.edu/~peters/courses/logsys/low-skol.htm
>>>
> =================
> Copy of my "lost?" note to Marc (Bov.3 - 6:59AM):
>
> Marc,
>
> I do not argue with 'your half' of the 'answer' you gave to the
> conference
> announcement of Jürgen Schm , I just ask for the 'other part': what
> should
> we call "a computer"?
>
> 'Anything' doing Comp? (meaning: whatever is doing it)?
>
> Will the conference be limited to that technically embryonic gadget -
> maybe
> even on a binary bases - we use with that limited software-input in
> 2006? a
> Turing machine?
>
> John M
>
> =========================================
>
>
>
> >
>
>
http://iridia.ulb.ac.be/~marchal/
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Received on Mon Nov 06 2006 - 06:57:02 PST