On 5/27/2011 10:18, andrel wrote:
> On 24-5-2011 11:22, Invisible wrote:
>
>> As I understand it, Gödel's theorem just says that everything is
>> impossible. That doesn't sound especially interesting.
>
> As I understand it it simply says that if you have a language that allo
ws
> you to make any interesting statement (like: all X have property Y)

Actually, the statement you need to be able to make is "some X have prope
rty 
Y, some don't."

"All X have Y" is not an interesting statement. Just like the halting 
problem is trivial for any language in which all programs halt, or no 
programs halt.

> It is interesting for at least two reason:
> 1) the way he proved it
> 2) it proved that all work done for centuries by the best mathematician
s to
> come to a simple and consistent set of axioms was useless.

Actually, that would be "consistent and complete". :-)

What *I* find fascinating is that there are actually mathematical problem
s 
that are Godel statements that aren't designed to be. I.e., it actually 
comes up in practice that you run across statements while investigating a
 
mathematical topic that you can't prove, but if you use a more powerful 
system, it turns out you can prove they're true. There's one that's along
 
the lines of the old riddle "if an odd number of people shake hands an od
d 
number of times..." but expanded to an infinite number of people, which y
ou 
can prove is true but unprovable.

-- 
Darren New, San Diego CA, USA (PST)
   "Coding without comments is like
    driving without turn signals."

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