Re: Theorem or Paradox

[Robert Seely <rags@math.mcgill.ca>]

It would take a book-length response to adequately reply to your
question - more about that later.  But in short here's an answer: a
theorem is the conclusion of a valid argument (i.e. a "proof") based
on certain assumptions ("hypotheses" or "axioms").  And Godel's result
is such a theorem.  I won't attempt to define a paradox, but certainly
it seems his theorem might also be regarded (and has been) as a
paradox as well.  But a better answer might be found in the book
"Godel's Theorem: an incomplete guide to its use and abuse", by Torkel
Franz\'en - I hugely recommend it, if you haven't already done so.

   -= rags =-

PS: Godel's theorem isn't based on the liar paradox - that's Tarski's
theorem - but rather on a closely related paradox about provability -
which isn't really a paradox after all ...

PPS - you don't really think theorems "close development or debate",
now, do you?!  Experience suggests otherwise I think.

On Sun, 8 Jan 2017, Patrik Eklund wrote:

> Since the Incompleteness Theorem uses the Liar Paradox, why is it called
> the Incompleteness Theorem and not the Incompleteness Paradox?
>
> A Theorem closes a development or debate, and calls for admiration
> (because the inventor did something supposedly good), whereas a Paradox
> opens up development and debate (since the detector has pointed at
> something being wrong), and delays the call for admiration of the
> disruptively innovative solution until it is really deserved.
>
> Best,
>
> Patrik
>
>
>
>

-- 
<rags@math.mcgill.ca>
<www.math.mcgill.ca/rags>


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]