[FoRK] Gödel's theorem
Albert S.
albert.scherbinsky
Mon Aug 15 19:46:43 PDT 2005
I think we have come full circle, back to
epistemology. To the extent that completeness is one
goal in the quest for Truth, we can say that goal will
never be reached. Mathematics may be nothing but the
Truth, but it sure ain't the whole Truth, so helped us
G?del.
--- Russell Turpin <deafbox at hotmail.com> wrote:
> Corinna:
> >What I'm getting at is if mathemeticians can
> (historically, that is -- I'm
> >not sure if they still are) be deeply and
> profoundly disturbed at the mere
> >thought that no formal system can contain all true
> statements about itself
> >(at least, I think that's what Godel implies...
>
> I'm not sure what you mean by "contained in."
> G?del's
> theorem about the incompleteness of arithmetic says
> any finite formalization of arithmetic fails to
> PROVE
> some of the truths of arithmetic. Second-order
> arithmetic "contains" all the truths, in the sense
> that
> it has only one model, up to isomorphism.
>
> >.. then doesn't it lend weight to metaphysics ..
>
> Not that I can see.
>
> >.. to suppose that there is truth (little t)
> outside of
> >known reality?
>
> Now, what does that have to do with metaphysics?
> There are many things we don't know, and that we
> won't ever know. G?del's theorem surprised only in
> proving that some of mathematics also includes
> things that never will be known. (a) If you think
> about it, the fact of that may be less surprising
> than
> that it can be rigorously proven. (b) Not everything
> is as deep as numbers. There are many kinds of
> mathematics that are complete. Euclidean geometry.
> Real algebra.
>
>
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