Impossible Set

Let R be the set of all sets which are not members of themselves. Then R is neither a member of itself nor not a member of itself. Aka RussellParadox. The interesting thing about this guy is that it undermines some fundamental mathematical assumptions.

Not really - you can prove anything from a false starting-point. Consider also 'This sentence is not true.'.

The point here is that "set of all sets which are not members of themselves" is an invalid concept. "Sets can contain anything" means "sets can contain any existing thing", not "sets can contain any describable thing". -- DanielKnapp

But this rule was created, as a patch, after Russell presented his paradox.

Well, we want to be able to define sets as entities that can contain anything. This sort of forces the common terminology of "class of all sets" rather than "SetOfAllSets".

Bull. "Set of all sets" is commonly used and makes perfect sense. Does it contain itself? Well, is it a set? Yes. So it contains itself. This means that its complete contents are infinite in size; so?

So, you seem not to have considered the set containing each set which doesn't contain itself, the subject of this page. You can't validly claim 'bull' for one but not the other since both lead to a contradiction.

Set of all sets does not lead to a contradiction. Yes it does - for example, what are you left with if you remove any sets which contain themselves?

What makes you think you can remove "any sets which contain themselves"? You get a contradiction if you assume that there's a set of all sets and that you can take a set, remove all elements of it satisfying any description at all, and have the result still be a set. If you diagnose the trouble as stemming from the first part, you get ZermeloFraenkel? set theory or some variant of it. If you blame the second part, you get Quine's NewFoundations or something of the sort. You can build a viable SetTheory either way.

So if you follow Quine and get a set of all sets, does it contain all ZermeloFraenkel? sets?

That's not entirely a meaningful question, but in so far as it has an answer the answer is no. It's hard to be more precise. (For me at the moment, anyway; perhaps someone can come up with a description-of-a-set that's instantiated in ZermeloFraenkel? but not in NewFoundations and where it's sensible to say it's the same set that exists in one theory but not in the other.)

Does either of these theories have a model within the other? If so, ...

It is not known that either theory has a model within the other. I think NFU may have a model within ZermeloFraenkel?. NFU is "NewFoundations with Urelemente", which means that as well as sets you can have Other Things with no elements. (If I were choosing the terminology, I'd call them "atoms".)

Upon reading this definition, I quickly result in R = U as my internal reasoning is unable to insert any set into itself (Exception: stack overflow by induction). - Joshua

See SetOfAllSets, UniversalSet, RussellParadox.

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