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Hi David. I'm sure you reached a satisfactory conclusion and certainly don't intend to discuss this particular topic any further, but use it to move on. I do hope you read this as I note your research is in set theory and the title of your article here includes the word "paradox".

There's also a famous paradox in set theory known as the Russell paradox (and its equally famous "barber" version) and it's my contention that this too is really an order of operations of operations problem just like the one here, and could just as easily be resolved with brackets, conventions like PEMDAS, or some other order marker.

In fact this paradox has little to do specifically with sets at all. It works for just about any other subject-verb- object formation, for example: man-shave-man, set-belong to-set, dog-eat-dog, including those occasions when that object is the reflexive pronoun him-/her-/itself. I suggest that the confusion arises not with the nature of the grammatical subject but with the tense of that verb. It arises from failure to recognise that each successive appearance of the verb in the argument that sets out the paradox is logically, if not grammatically, a different tense and refers to a different time. The paradox disappears when we make that difference explicit, when we differentiate with respect to logical time. The barber shaves today all and only those people who didn't shave themselves yesterday, so if he didn't shave himself yesterday, he does today. The Russell set includes all and only those sets which didn't previously include themselves. So if it didn't include itself before it does now, but if it did before then it doesn't now.

In math when faced with the contradictory result arising from failure to put explicitly different relative times on the division, multiplication, and addition operations in an expression like 6 ÷ 2(1 + 2) = 4 we eventually find a way of nailing them down, so too with the operations of set inclusion and exclusion, or shaving and not shaving, or whatever.

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