Let S be the set S={X: X is not an element of itself}. Then assume S is an element of S. If so, S is not an element of itself - a contradiction. Assume S is not an element of S. Then S is an element of itself - another contradiction.

heh, wondered how long it would take Russell's paradox to appear - but 3 mins - that's impressive!

@Thuc - the second doesn't work, since you might be by the time I read it.

It does! But what matters is that he said it when he wasn't dead!

Well yes it does, my point is it doesn't always :P

I always lie.
Do not read this.
Absolutes do not exist.
There is always an exception to any rule.
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^^ That last one really annoys me - you see it in exam papers all the time!

"The statement below is false."
"The statement above is true."

e. All of the above

The value cola tasted good.

A = B if and only if A = C and only C. And B does not equal C

My favourite is:

"If this sentence is true, Santa Claus exists."

But that's not paradoxical. It's false since Santa Claus doesn't exist. Proof by contrapositive.

Oh...I get it.

@CM: this can be true, but it implies that = is not transitive. (which would be unusual)

ie A = B and B = C then this implies A = C

@ConMan:
A => !A

i don't get the santa clause.

@ Ora: I said that B does not equal C.

*groan*

@ConMan - he was explaining what transitivity is

I just looked up a lot of advanced algebra. Equality is a transitive relation so your statement is not paradoxical at all. It's just wrong (sorry!)

I won the game

The santa claus one's tricky to understand.
I'll try to outline the steps.
STATEMENT
"If this sentence is true, Santa Claus exists."

1. Agree that IF the statement was true, then Santa Claus would exist.
I think this must be true, as IF the statement if correct in saying that if the statement's correct, santa claus exists, and it is indeed correct, then SC must exist.

2. Agree that the statement is true when it says that IF the statement is true, SC exists.
In part 1, you agreed that IF the statement is true, SC exists. The statement states exactly that, so it is true.

3. As you've agreed in part 2 that the statement is indeed true, and in part 1 agreed that IF the statement's true, SA exists, SA does exist.

that's nice.

Though i prefer to view it as

'unknown condition => Santa Claus exists.'

as self-referential statement i can't evaluate the truth value of the sentence.

right? you still lost me once or twice in the logic though...

oh and at algebra, i was assuming the = could be replaced by undefined relationship ~

A relates to B: written A ~ B iff A ~ C and only C AND B !~ C

So if ~ is transitive (ie A ~ B AND B ~ C implies A ~ C ;like the less than relationship)
and symmetric (ie A ~ B implies B ~ A) then A ~ B iff A ~ C

means we have first A ~ B => B ~ A => B ~ C => C ~ B which contradicts the second part of the if and only if statement.

However should we take away the transitiveness of the relationship ~ that logic falls down.

I still don't get it... Santa's existence is part of the sentence, therefore the sentence is false because Santa doesn't exist. He didn't say some statement was true, but the whole sentence. Gramatically the sentence is false and not a paradox.

What is so paradoxical about e. All of the above? Had it been preceded by d. None of the above, I would agree, but the statement in and of itself is not a paradox. The paradox is in the details, people. Details, details, details!

But the sentence is true when it says that if its true, SC exists, would you not agree?

The bit about santa claus is not important. The paradox is that the sentence's logic can be used to prove any conclusion

rlumley is a good strategist.

The sentence is false, therefore no paradox. I see what you are getting at, but the entire sentence is patently false. Do you deny that the sentence is false? I mean the *entire* sentence because that is what was specified in the IF clause.