On 5/9/2012 1:21 AM, Invisible wrote:
>> 1) Does the set of all sets that do not contain their own complement
>> contain itself?
>>
>> There is no such set, and thus the statement is vacuously true.
>
> I'm not sure how you came to the conclusion that no such set exists.
>
> Since /by definition/ a set can never contain its own complement (that's
> what I complement /is/), the set above contains all possible sets -
> including itself.

Right, and such a "set of all sets" does not exist within standard set 
theory.  It appears, however, that you are unfamiliar with standard 
(ZFC) set theory, the your misconception is understandable.

I am, by the way, glossing over your use of "compliment" which strictly 
speaking isn't well defined the way you've used it.

>> 2) Now tell me, is the set of all sets that list themselves listed in
>> itself?
>>
>> There are no sets which contain themselves, thus the set of all these
>> sets it in fact the empty set. The empty set does not contain itself, so
>> the answer is "no".
 >
> Again, I'm not sure how you come to the conclusion that a set cannot
> contain itself.

Oh come on, it's one of the axioms of ZFC set theory! 
http://en.wikipedia.org/wiki/Axiom_of_regularity

> Even if that were true, it would imply that the set in
> question /does not exist/, rather than that it is empty.

Which is exactly what I said, you'll notice.  I'll quote to save you the 
trouble of reading: "thus the set of all these sets it in fact the empty 
set".

> The answer of course is that the definition of the set is inconsistent.
> If the set does not contain itself, that implies that it does contain
> itself, and vice versa. So the answer is neither "yes" nor "no". The
> answer is "undefined".

That was the answer at around 1901 before modern set theory existed, and 
it was sufficient to destroy the notion of "set" which led to it.  You 
appear to be using a notion of "set theory" like this which hasn't 
really been popular for over a century.  In ZFC set theory there is no 
paradox because you can't create the sort of sets needed for the paradox 
to happen.

Post a reply to this message