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From: spudnik on 21 Jan 2010 13:55
kind of academic,
not to actually work a problem in (say) numbertheory,
such as Fermat's proof by "infinite descent
for n=4 in x^n + y^n = z^n --
the only reuquired non-odd exponent!

--les OUevre!
http://wlym.com
From: KevinSimonson on 21 Jan 2010 15:14
On Jan 20, 5:17 pm, Joshua Cranmer <Pidgeo...(a)verizon.invalid> wrote:

=Assume T is inconsistent. If it is inconsistent, then one can derive
=a contradiction. A contradiction is by definition not true, so
=therefore we have proved our assumption to be false, and T must be
=consistent.
=
=Ultimately, a proof by contradiction assumes that a contradiction
=cannot be proved in said formal system. You assumed that a
=contradiction exists in T, which renders invalid a proof by
=contradiction.

Thanks, Joshua; this is precisely the answer I was looking for. It
definitely gives me something to think about, anyhow. I note, with a
big sigh of relief, that standard math may not be inconsistent after
all!

Kevin Simonson

"You'll never get to heaven, or even to LA,
if you don't believe there's a way."
from _Why Not_


From: Marshall on 21 Jan 2010 23:19
On Jan 21, 12:14 pm, KevinSimonson <kvnsm...(a)hotmail.com> wrote:
> On Jan 20, 5:17 pm, Joshua Cranmer <Pidgeo...(a)verizon.invalid> wrote:
>
> =Assume T is inconsistent. If it is inconsistent, then one can derive
> =a contradiction. A contradiction is by definition not true, so
> =therefore we have proved our assumption to be false, and T must be
> =consistent.
> =
> =Ultimately, a proof by contradiction assumes that a contradiction
> =cannot be proved in said formal system. You assumed that a
> =contradiction exists in T, which renders invalid a proof by
> =contradiction.
>
> Thanks, Joshua; this is precisely the answer I was looking for.  It
> definitely gives me something to think about, anyhow.  I note, with a
> big sigh of relief, that standard math may not be inconsistent after
> all!

"For every complex problem there is an answer that is clear, simple,
and wrong."

-- H. L. Mencken


You would do well to pay more attention to the replies
that have less of the wrong in them, even if it means
they also have less of the clear and simple. Look to
Hughes, Koskensilta, and Ullrich, for example.


Marshall
From: Andrew Usher on 24 Jan 2010 19:25
On Jan 21, 6:54 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:

> Rules of inference are valid or not independently of the theory in
> which they are used.  The definition of validity does not refer to the
> theory.  

Well, not my definition. If you say so, then your logic is just
meaningless symbol manipulation.

Andrew Usher
From: Andrew Usher on 24 Jan 2010 19:26
On Jan 21, 7:57 am, David C. Ullrich <ullr...(a)math.okstate.edu> wrote:

> >Only in a vacuous sense. Mathematicians do assume 'P xor not P'
> >because it is true, that is, true in real, informal logic. The fact
> >that Goedel's theorem shows that it is not always so in any formal
> >system
>
> For heaven's sake, where did you get the idea that Godel's
> theorem says that "P xoe not P" is not always so in any
> formal system?

It shows that it isn't always provable, which in the context of
rigorous proof means the same thing.

Andrew Usher
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