This is the first time I've seen this paradox. It is confusing. But it just seems like a play on words. You can't seem to cross more than two levels unless you change the definition of the set. If A=odd #s and B=even #'s then A and B are not members of themselves. So if C=A+B then C could be considered the set of sets that are not members of themselves, or C could be the set of all real integers. But if A was the whole number digits of pi and B was all prime #'s then you can't really define A+B. I guess it is in the hierarchy. You can't really cross two levels by defining the second level as C when it is really undefined. You're just puting a label on something that doesn't fit. It is like saying A is all red motorcycles and B is all blue motorcycles and calling the set A+B the set of motorcycles. It's not true. You either would have to add them together and say the set of all blue and red motorcycles, or keep it as 2 separate sets that can not be combined because there is no definition to that set. So by saying you have a set of sets that are not members of themselves, you could be talking about a set of infinite possibilities with no real parameter defined. I guess in my opinion when you say a set of sets that are not a member of themselves, you could be talking about anything. Now if that "anything" happens to be a group of things that have a physical rule that each individual thing shares then it could be a set defined by that rule. But, if there is no physical common bond other than the "anything" is just a group of sets drawn together and labeled with a variable, then in my opinion they are not really a set. More of an irrational set. And in the Barber Paradox it seems to me that this is just an irrational statement. I shave everyone that doesn't shave themselves. Well, he misspoke. He shaves everyone who doesn't shave themselves, other than himself. The set was not defined properly. Basically, the barber is the definition of the set. The flaw to me is this. People who do not shave themselves are individuals with a rule. The barber is really a parameter, not an individual. You must first be an individual then follow the rule to be in the group. How can a set be a member of individual motorcycles that are red? It's not a motorcycle, it is a set. Anyway, that's my thoughts on the subject. I guess I would have to see how this applies in mathematics to fully understand.
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