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:
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).
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
> Now "uncountability" is not an absolute notion (that is the
> Lowenheim-Skolem lesson).
> Careful: uncomputability is absolute.
> 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:
>> "[EMAIL PROTECTED]" <[EMAIL PROTECTED]>:
>>> Ah the famous Juergen Schmidhuber! :)
>>> Is the universe a computer. Well, if you define 'universe' to mean
>>> 'everything which exists' and you're a mathematical platonist and
>>> reality to infinite sets and uncomputables, the answer must be NO,
>>> since if uncomputable numbers are objectively real (strong platonism)
>>> they are 'things' and therefore 'part of the universe' which are by
>>> definition not computable.
>>> But if by 'universe' you just mean 'physical reality' or 'discrete
>>> mathematics' or you refuse to grant platonic reality to uncomputables
>>> or infinite sets (anti-platonism or weaker platonism) then the answer
>>> could be YES, the universe is a computer.
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