Le 06-nov.-06, à 03:46, [EMAIL PROTECTED] wrote:
> Bruno Marchal wrote:
>> 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.
> Well, 'existence' would certainly be a stronger notion of platonism
> than mere 'definability'.
> So Bruno, what would your answer be to the question of whether the
> universe is a computer or not? I think it all depends on how you
> define 'universe' and 'computer' ;)
> Personally, my answer is no, I don't think the universe is a computer.
> I define 'universe' to mean 'everything which exists' and computer to
> mean 'anything which is Turing computable'. Since I think
> uncomputables do exist, they are part of the universe and they are not
> Turing computable so the universe as a whole can't be a computer.
I agree with you. Even the "seemingly" tiny "universe of numbers" is
full of non computable stuff.
Recall that Church thesis can be used to prove the existence of non
computable objects in very few lines (as I have done more or less
recently in posts to John and Tom).
> But one doesn't need to believe in uncomputables to doubt that the
> universe as a whole is a computer. There is also the problem of
> infinite quantities to contend with. Something which is computable is
> most likely finite (by holographic string principles), but if there
> exist things with infinite extent or quntification (like space for
> instance) it's hard to see how the universe as a whole could be defined
> as a computer.
Hmmm... The very notion of general computability needs the infinite
(the finite realm is *trivially* (obviously) computable). So "infinite"
per se is not directly responsible of the non computability. It is the
diagonalization closure of the computable realm (I can come back on
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