Re: Concept_sitting

in the domain of languages,
which problem (s) could result from a statement refering to itself ?
maybe it's a naive question but i'd like to know your opinon.
ebs

Le 16 janv. 09 à 13:19, Mark J. Reed a écrit :

> On Thu, Jan 15, 2009 at 10:22 PM, Sai Emrys <saizai@...> wrote:
>> Gödel, for example, proved that any (mathematical) system necessarily
>> has certain axioms that cannot be proved within that system.
>
> No, he didn't.  If a  mathematical system that uses "axioms", those
> are automatically not provable within the system, because that's what
> axioms are: the basic statements you accept without proof.
>
> What Gödel demonstrated was that, past a certainly surprisingly low
> level of complexity, you can't create a system of symbolic logic with
> no paradoxes.  He was refuting the work of others who believed that it
> was possible to extend the basic Euclidian idea of "start with a few
> axioms and derive theorems from them according to fixed rules of
> logic" until you could define all of mathematics in a symbolic system
> where all true statements were provably true and all false statements
> were provably false.  These guys (Whitehead and Russel being the major
> work) took nothing for granted, even 1 + 1 = 2 was a long proof in the
> logic system.
>
> But any system complex enough to tackle that job will inherently allow
> the creation of statements that are not provably true or false.  They
> are analogous to sentences of the "this statement is false" variety.
> W&R tried to dismiss them out of hand, but Goedel showed (by providing
> a general method that always works) that no matter how carefully you
> try to restrict the domain of reference (so that statements can't
> refer to themselves, for instance), there's always a way to generate
> such a paradox.
>
> --
> Mark J. Reed <markjreed@...>

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