> 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.

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

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".

>> Interestingly, in Portal 2 GLaDDOS poses nearly the exact same question
>> to Wheatly, who also responds with "false".
>
> Maybe an in-joke by some math-savvy writers? Assuming my reasoning is
> correct that is.

The actual statement was the liar paradox, to which Wheatly, being 
scientifically engineered to be the greatest moron ever, replies

"Er... false. Yes, definitely false. Although I think I may have heard 
this one before, to be fair."

The exasperated GLaDDOS retorts "There IS no answer, you idiot! It's a 
paradox!!"

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