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[Graystripe77]

This statement is both true and false, and also neither.

Does anyone have any others?

[timchen]

I am telling a lie.

[Ozle]

The Unexpected Hanging Paradox.

Judge tells man he will be hanged one day next week, but that it will be a surprise.
Man goes away happy because he knows he cannot be hanged now

He reasons that he cannot be hung on friday, because if it gets to thursday evening and he is not hung, then the hanging must be on Friday, and this wouldnt be a surprise, so it cant be.
Once he has ruled out Friday for definite, he then also rules out Thursday. Because if it gets to wednesday night he will know he is be hanged on thursday, which wouldnt be a surprise.
Following this process rules out all the days left in the week. So he goes back to his cell happy he cannot be hung.

Judge hangs him on Wednesday which was a complete surprise to him!

[jotheonah]

Is the set of sets which are not members of themselves a member of itself?

Or, put another way, a town has a barber who shaves all the men in town who don't shave themselves, and only the men in town who don't shave themselves. Does he shave himself?

[O]

Is the set of sets which are not members of themselves a member of itself?

There is no such set. Mathematicians changed the definition of set as to avoid that issue.   

[ycz6]

http://en.wikipedia.org/wiki/List_of_paradoxes

Have fun! Curry's Paradox is my favorite.

[jotheonah]

Is the set of sets which are not members of themselves a member of itself?

There is no such set. Mathematicians changed the definition of set as to avoid that issue.   

Bertrand Russel is such a badass he broke math, and they had to change it to make it work again.

For an interesting book about paradoxes (and everything else) I would recommend Godel, Escher, Bach: An Eternal Golden Braid by Douglas Hoftadter. It's fun.

[Deadlock39]

Is the set of sets which are not members of themselves a member of itself?

Or, put another way, a town has a barber who shaves all the men in town who don't shave themselves, and only the men in town who don't shave themselves. Does he shave himself?

The barber is a woman

[eHalcyon]

Is the set of sets which are not members of themselves a member of itself?

Or, put another way, a town has a barber who shaves all the men in town who don't shave themselves, and only the men in town who don't shave themselves. Does he shave himself?

The barber is a woman

I had this exact thought but decided that it doesn't work because the problem asks "does he shave himself?", thus implying that the barber is a man. 

[DStu]

For an interesting book about paradoxes (and everything else) I would recommend Godel, Escher, Bach: An Eternal Golden Braid by Douglas Hoftadter. It's fun.

... for a suitable definition of "fun"...

[jotheonah]

For an interesting book about paradoxes (and everything else) I would recommend Godel, Escher, Bach: An Eternal Golden Braid by Douglas Hoftadter. It's fun.

... for a suitable definition of "fun"...

Eh, it's either your thing or it isn't. You'll know pretty quickly.

[ehunt]

This one bugged me a lot recently. An evil wizard plays a game. Everyone knows the rules of the game. It goes like this.

He picks a person at random and calls him into a dark room. He rolls a six sided die. If he rolls six, he kills that person and the game is over forever. If he rolls less than six, he sets that person free, and never bothers that particular person again, but the game continues.

If the game continues, then instead of calling in one person, the evil wizard calls in nine. He rolls a six sided die. If he rolls six, he kills all nine and the game is over forever. If he rolls less than six, he sets all nine free, and never bothers those particular nine people again, but the game continues.

If the game continues, then instead of calling in nine people, he calls in ninety people ...

The game continues in this manner until he rolls a six and the game ends (and everyone in the room is slaughtered). (The pattern for the number of people he calls into the room is: 1, 9, 90, 900, 9000, ... etc. The thing to note is that at any time after the first stage, the number of people in the room is equal to 90% of the number of people who have ever been in the room.)

Now, here's the deal. You get a call from a stranger informing you that your friend Julian entered the room. The stranger doesn't know what happened after that. He also knows the game is over, but he doesn't know when it ended. What is the probability that Julian is dead?

The paradox: there are two good answers.

Answer one: The probability is 1 in 6. We know that Julian entered the room with some number, possibly zero, of other people. At that time, the wizard rolled a die, and there was a one in six chance that he killed Julian and his cellmates.

Answer two: The probability is 9 in 10. We know that Julian was in the room, and nine in ten people who ever entered the room are dead.

[Axxle]

Is the set of sets which are not members of themselves a member of itself?

Or, put another way, a town has a barber who shaves all the men in town who don't shave themselves, and only the men in town who don't shave themselves. Does he shave himself?

The barber is a woman

I had this exact thought but decided that it doesn't work because the problem asks "does he shave himself?", thus implying that the barber is a man. 

The barber is a boy

[Axxle]

This one bugged me a lot recently. An evil wizard plays a game. Everyone knows the rules of the game. It goes like this.

He picks a person at random and calls him into a dark room. He rolls a six sided die. If he rolls six, he kills that person and the game is over forever. If he rolls less than six, he sets that person free, and never bothers that particular person again, but the game continues.

If the game continues, then instead of calling in one person, the evil wizard calls in nine. He rolls a six sided die. If he rolls six, he kills all nine and the game is over forever. If he rolls less than six, he sets all nine free, and never bothers those particular nine people again, but the game continues.

If the game continues, then instead of calling in nine people, he calls in ninety people ...

The game continues in this manner until he rolls a six and the game ends (and everyone in the room is slaughtered). (The pattern for the number of people he calls into the room is: 1, 9, 90, 900, 9000, ... etc. The thing to note is that at any time after the first stage, the number of people in the room is equal to 90% of the number of people who have ever been in the room.)

Now, here's the deal. You get a call from a stranger informing you that your friend Julian entered the room. The stranger doesn't know what happened after that. He also knows the game is over, but he doesn't know when it ended. What is the probability that Julian is dead?

The paradox: there are two good answers.

Answer one: The probability is 1 in 6. We know that Julian entered the room with some number, possibly zero, of other people. At that time, the wizard rolled a die, and there was a one in six chance that he killed Julian and his cellmates.

Answer two: The probability is 9 in 10. We know that Julian was in the room, and nine in ten people who ever entered the room are dead.

Actually, I believe that Answer two is the only right answer since we know that the game is over.  We're asking the probability of her dying out of X number of people, not what are her chances of dying given that she's up for the roll.  Hmm... I'm explaining this poorly but xkcd explained it some here: http://forums.xkcd.com/viewtopic.php?t=81

[Tables]

Is the set of sets which are not members of themselves a member of itself?

There is no such set. Mathematicians changed the definition of set as to avoid that issue.   

Unless you ask my combinatorics lecturer, in which case there is no definition for a set (but there is for a multiset, and it depends on the definition of a set...)