Re: CHAT: mathematics

On Sat, 18 Nov 2000, John Cowan wrote:

> On Sat, 18 Nov 2000, Yoon Ha Lee wrote:
>
> > I love how the history of mathematical prejudice is recorded in the names
> > of number-types: rational, irrational, imaginary, complex, nonstandard...<G>
>
> Don't forget negative.
I think someone else listed a bunch of others.  But hey, that's what the
ellipsis was for, right?  =^)

> > > Wouldn't it be cool if there was a finite proof for G?  Nobody actually
> > > knows if it's true -- but if it were, nonstandard numbers would be
> > > *hard-wired* into number theory, willy-nilly.
> >
> > G?  <puzzled look>  Clarify, please?  I'm mightily curious, but also
> > rather ignorant.  :-(  (The only thing that comes to mind is "g" in
> > psychology, which I'm guessing is an entirely different animal.)
>
> Right.  G is the Goedel sentence, the one whose meta-mathematical
> interpretation is "G has no proof".  (Its purely *mathematical*
> interpretation is just a relationship between some rather large
> numbers, but when you map the numbers onto statements in proofs,
> you find that G asserts that it has no proof.)
>
> However, I blundered above by saying "finite proof for G"; it's
> precisely Goedel's Theorem that G has no proof. I meant "finite
> proof for not-G".  This is an open question, and most number theory people
> believe it's false, but there is no proof either way.
Hmm.  I've been introduced to Gödel's theorem (or one of them?) on three
separate occasions (a comp sci class, a math lecture series, and reading
_Gödel, Escher, Bach_) but I can't say that the first two gave much
detail, and as for _GEB_, I can't claim to have understood much of it (I
read it during HS).  Fascinating stuff--I've been following the messages
this thread has generated but confess I'm getting lost.  :-p  Oh well, I
am only a math major, not a mathematician...

YHL