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 ) 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
> 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:
>>> 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
>>> grant
>>> 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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