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
>
>
> "marc.geddes.domain.name.hidden" <marc.geddes.domain.name.hidden.com>:
>
>>
>> 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.
http://iridia.ulb.ac.be/~marchal/
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Received on Fri Nov 03 2006 - 07:51:38 PST