Re: Concept_sitting

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@...>

Reply