The late Groucho Marx said words to these effects: " …I wouln’t join a club that would have ME as a member!" Can a part of a set belong to another set (which excludes the properties of the first set? Any mathematicians here? :smack:
I think it just indicates that the set of all clubs that Groucho will join is empty. Either a club will have him as a member (in which case Groucho won’t join it) or it will not (in which case Groucho can’t join it).
More paradoxical is an idea I came across in a New Scientist article about the Hilbert Hotel (a hotel with an infinite number of rooms). Guests can join any number of clubs that meet in the hotel, and all clubs meet in someone’s room. The Absentees’ Club is the club for all people who do not belong to the club that meets in their room. But what about the man whose room the AC meets in? Either he belongs to it (in which case he does belong to the club that meets in his room) or he doesn’t (in which case the club doesn’t include everyone who don’t belong to the club that meets in their room) - in other words, the Absentees’ Club, despite its definition, must either include an ineligible member or exclude an eligible one. 
I’m not a mathematician, but I’d say the answer depends on the interpretation of “part” and “excludes”. Is “excludes” taken to mean “logically contradictory” or just “different”? Is “part” taken to mean “subset”?
If I’m reading you right, you’re asking whether any subset of {x | ~P(x)} can be a subset of {x | P(x)}. It’s not possible. What that has to do with Groucho’s jokes is a little less clear.
This is a variant on Russell’s paradox, sometimes called the Barber paradox.
The Absentees’ Club should meet in the lobby. Too bad that can’t be transferred into set theory.
It’s not really a paradox of any kind because he’s proposing a hypothetical. He’s not really declaring that he’s a member of a club that excludes him.
To answer your question literally, Marx’s joke doesn’t express any axiom or principle of set theory. It can be expressed “set theoretically” however, as a claim that:
Where g is Groucho Marx,
and where C1, C2, and so on, are the sets such that Cn consists in all of the things things which would be admitted as a member of club n (assume we have placed all the clubs that exist in some kind of order already),
and where D is the set of all of C1, C2, and so on which are such that g would join them if given the opportunity,
For all n, if g is an element of Cn, then Cn is not an element of D.
I don’t see any inconsistency or paradox in the claim that these sets can exist together. If g is in a Cn, then that Cn is not in D. Meanwhile, if g is not in a Cn, then that Cn may or may not be in D. I don’t see any contradiction that can be drawn from these facts.
-Kris
In other words, basically, what Liberal said. 
-FrL-
That should read “…g would join club n…”
You don’t get to decide which sets you are a member of.
-FrL-
This was interesting. I can tell I’m an engineer, not a mathematician or logician, because my first thought was that the wording should have been:
I guess this means that I didn’t really get the significance of Russell’s paradox.
What about the empty set? 
I would think it means you understood the significance of the paradox right off the bat, and took steps to avoid running into it.
-FrL-
The barber paradox seems pretty obvious to me in that it is not the same shaving someone as a barber shaves a customer than shaving yourself. The barber shaves himself as a person, not as a barber (heck, even the technique is different)
Here’s the paradox as written at the wiki site:
So the question is, is it possible for someone to follow a rule that says:
“Shave all and only those who do not shave themselves.”
The point is its impossible to follow that rule. If you shave yourself, then you are, according to the rule, not supposed to shave yourself. But if you don’t shave yourself, you are, according to the rule, supposed to shave yourself.
So even though the rule appears at first to make sense and to be one one could plausibly follow, it turns out that it is impossible to follow the rule, and so it would be incorrect to claim that the rule actually governs anyone’s behavior.
If I am reading you correctly, you are saying the barber can follow this rule:
“Shave all and only those other than yourself who do not shave themselves.”
This is indeed a rule it is possible to follow. In fact, it’s the rule that is probably intended by anyone who says “shave all and only those who do not shave themselves.” But this new rule is, strictly speaking, a different rule than the one that is given in the formulation of the paradox. The question is why that rule can’t be followed, and what should be done about the fact that it can’t be followed.
To the first question, you seem to be giving the answer “The rule can be followed, because shaving yourself is a different kind of action than shaving someone else. So when the rule says ‘shave’ its referring to shaving someone else, not shaving oneself.”
I don’t think that’s adequate. Every act of shaving is in some way distinct from every other act of shaving. But if you told me that I did not in fact perform an act of shaving this morning (I shaved myself) you would be wrong. Someone shaving himself, and someone shaving someone else, are both performing an act of shaving. The rule in question makes explicit reference only to the performance of acts of shaving.
So, again, while in normal English we probably all would understand someone giving that rule to mean the non-self-shaving version of the rule, nevertheles, the fact is, the rule itself doesn’t say that self-shaving is excluded from its scope. It includes all acts of shaving within its scope.
So the question is, when we read the rule strictly according to its literal meaning, we end up with a rule it is impossible to follow, and we’d like to know why it is impossible, and what should be done about its being impossible.
-FrL-
Now that the question has been answered, I think Curry’s paradox is more interesting than any of these because it is extremely simple and requires no negation:
“If this statement is true, then God exists.”
Following standard methods of deduction, we assume that the statement is true and derive from the conditional that God exists. That means that the statement is indeed true. So if the statement is indeed true, then God indeed exists because the statement says He does. But the beauty of it is why the statement is a paradox. It is because any predicate will produce the same truth value, including a predicate that directly contradicts:
“It this statement is true, then this statement is false.”
So far, I’ve not seen anyone satisfactorily explain it away.
The significance was that, in the early 1900’s, the paradox was a “valid” way of defining a set by the rules of set theory at the time. The solution, of course, was to change the rules of set theory so that such a construction was invalid, but doing so was easier said than done.
The problem there is the construction “this statement”. It is very difficult to construct, in any formal logical system, a self-referential statement. Depending on what you’re trying to make the statement say about itself, it can occasionally be possible, and Gödel’s famous Incompleteness Theorem is based on such a self-referential statement. But in many other cases, such as “If this statement is true, God exists”, or the even simpler “This statement is false”, there does not exist any way of formally constructing the statement.
Isn’t it rather that there exist many ways of formally constructing these kinds of statements, but that each of these ways leads to unhappy results?
-FrL-
Well, that’s true. Almost every natural language will accomodate the construction, but for logic self-reference more problematic. (For one thing, there’s the problem of circularity.) Godel had to invent his own self-referential language, and there’s nothing wrong with that.
But you can construct a so-called paraconsistent entailment semantics (which is what Graham Priest did ten years ago or so) that will almost resolve the paradox, except that there’s the problem of supervenience (his world is non-normal). One solution offered as recently as a couple years ago (by Beall) is simply to declare that logical laws fail by fiat. Since fiat is arbitrary, it demands a lack of supervenience. And so, problem solved. But that’s like eating air. Not very satisfying.
But now the pendulum is swinging back the other way. Greg Restall proved last year, in Curry’s Revenge: the costs of non-classical solutions to the paradoxes of self-reference (PDF), that non-classical solutions all come up short. You have to reject large disjunctions, the law of distribution, the transitivity of entailment (e.g. A -> B, B -> C means that A would not imply C), or reject the schema itself, which means rejecting the whole logic you’ve invented (thereby leaving only meaningless semantics).
For classical solutions, you need at least referential identity (something to mean “this element”), a law of non-contradiction, a law of identity, a rule of modus ponens, and a self-contained truth predicate (something that says an element is true in your language). Natural language has all these. (And that’s why it’s so easy to read and write the statement in English.) But mathematical languages generally don’t.
It’s really a booger.