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);
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) - 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.
John M
----- 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
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Received on Sat Nov 04 2006 - 18:48:00 PST