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Consider the 'set' behind Russell's Paradox:

R = { x | x is a set and xx }

in light of Cantor's definition of set ("aggregate"/Menge) in his CONTRIBUTIONS TO THE FOUNDING OF THE THEORY OF TRANSFINITE NUMBERS (Dover edition),

By an 'aggregate'...we are to understand any collection into a whole... M of definite and separate objects m of our intuition or our thought. These objects are called the 'elements' of M.

One should note that in this definition, 'objecthood' is primary.

Considering R once again, R certainly has elements and according to Cantor's definition of set can definitely be considered one. Let me now ask the question that leads us into Russell's Paradox:

"Is R a member of R?"

Because R has elements, it can definitely be considered a 'definite and separate object of our intuition or our thought' and as such can seemingly have certain attributes satisfying it and others not satisfying it.

Russell's Paradox is that assuming 'R is not a member of R' implies 'R is a member of R', which implies again 'R is not a member of R'.

Since R's 'objecthood' is primary, why doesn't it make sense to say that R can neither have the attributes '__ is a member of R', nor not-'__ is a member of R' correctly attributed to it? If this is the case then Russell's Paradox is dissolved, since it is the assumption that R must satisfy either '__ is a member of R' or not-'__ is a member of R' that seemingly gets us into the paradox to begin with.

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Not a direct answer, but you may find Leśniewski's mereology interesting as it is to a significant degree motivated by Russell's paradox (as well as Leśniewski's aversion to the empty set and, by virtue of his nominalism, his insistence that singletons are equal to their lone members i.e. x = {x}). –  danielm Nov 23 '12 at 21:59
@danielm: You are right--'naive' mereology is very applicable to Russell's paradox and the continuum as well. For example, assume the world is at base nothing but 'gunk' (i.e., 'gunk' is primary) and one constructs entities via their attributes out of the 'gunk'. Construct the Russell Set R from the 'gunk' and R can neither be nor not be a member of itself (i.e. those attributes cannot be attributed to R). Regarding the continuum, it cannot be defined as a set of points so the Continuum Hypothesis is false. –  Thomas Benjamin Dec 14 '12 at 6:33
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2 Answers

Since R's 'objecthood' is primary, why doesn't it make sense to say that R can neither have the attributes is a member of R nor not-is a member of R correctly attributed to it? If this is the case then Russell's Paradox is dissolved, since it is the assumption that R must satisfy either is a member of R or not-'is a member of R that seemingly gets us into the paradox to begin with.

If I understand your question correctly, you don't want to declare these membership relations to be undecidable statements (as the other answers seem to interpret your question); you want to restrict the collection R not to have a set membership relation.

Good news, you just discovered proper classes!

Leaving aside the remark about Cantor's "objecthood", which I could't really follow, your intuition leads you in the right direction.

Let's analyse the situation:

R = { x : x ∉ x } yielding R ∉ R ⇔ R ∈ R could be obtained in set theory because an informal unrestricted comprehension principle was being used. Hence, the problem can be dealt with by carefully restricting the comprehension principle. That's exactly how it's being done in contemporary set theory, using an axiom schema of restricted comprehension. The result is that ZFC doesn't allow to define R, or, to state it more exactly, R can be defined as a reductio ad absurdum to prove that the "set of all sets" doesn't exist, i.e. assuming its existence leads to the contradiction described by Russell's paradox.

The salient point, however, is that what the proof really tells us is that the set of all sets cannot be a set. In fact, Russell's paradox, as well as Cantor's paradox and Burali-Forti paradox all tell us simply that some collections, like "the set of all sets" are not sets. The father of set theory, Georg Cantor, thought of these collections, which he called "absolute infinities" as beyond the reach of mathematics and went mystical about them. As it turns out, this evaluation was too pessimistic. The kind of collections which Cantor called "absolute infinities" are known today as proper classes. (You may consult additionally this brief introduction).

Simply put, the concept of class can be introduced this way:

A class x is a set iff there is a class y such that x ∈ y. A class which is not a set is said to be a proper class.

Now, suppose that R ∉ R. If you suppose that R is a set, you get a contradiction, so R must be a proper class.

In ZFC we can talk about proper classes only informally. However, there are alternative foundational systems, known also as class theories, which - surprise! - allow to treat proper classes formally beside sets. The most "explicit" of these systems is Morse–Kelley set theory, which admits proper classes as basic object alongside with sets. But there are many other approaches.

See also:

Some notable proper classes:

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Okay, so I read everything after "Good news...!" in Professor Farnsworth's voice. –  Joseph Weissman Nov 23 '12 at 14:35
… For the uninitiated: Good news everyone! –  DBK Nov 23 '12 at 20:39
@DBK: What I am trying to say is that R is still a set, because it can still be a member of some class y--it just can neither be nor not be a member of itself--those attributes just don't apply to R. What Russell himself says is that R does not form a totality (which is of course correct assuming excluded middle). What I am saying is that classical logic may not be the proper logic for Naive set theory, just as classical logic may not be the proper logic for quantum systems. I hope this clarifies--not confuses. –  Thomas Benjamin Nov 24 '12 at 6:01
@Thomas: My bad. But what you are trying to say in set theoretic terms is that a certain set membership relation shouldn't selectively apply, namely between the set and itself. (It is still not clear if we ought to exclude every set from applying to itself or just R). This is a completely arbitrary and ad hoc restriction. Also, I don't understand what this has to do with LEM in general. But even assuming that it is somehow connected, that's still not a departure from classical logic because in all other membership relations LEM should still hold, following your line of thought. –  DBK Nov 24 '12 at 14:27
@Thomas: The closest "object" fulfilling your restriction I can think of is an urelement. These urelements are the "definite and separate objects" Cantor is referring to in his definition, but they are not sets to begin with. And an urelement U is actually defined as not having any elements, so - in your parlance - ∉ can still "be predicated" of U. –  DBK Nov 24 '12 at 14:55
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You have constructed R and now you ask whether R is a member of R. A positive or negative answer to this question leads to a contradiction. You are resolving it by saying R cannot have the properties: 'is a member of R' or not-'is a member of R'. This is absurd. It's like saying some statement cannot be 'true' or 'false'. Unless you are using some logic other than the standard one, no statement can be neither true nor false. (That is, every statement is true or false.)

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You seem to assume attributes are predicates - if they weren't why need we assume excluded middle for them? Suppose we say that each attribute R is associated with a predicate R', and then that R is an attribute of T when R'(T) is necessarily true. Then excluded middle doesn't hold for these kind of attributes in general. –  Charles Stewart Nov 22 '12 at 13:48
What about the Continuum Hypothesis by your namesake? I'm really curious, could it be that it is either true or false, but simply cannot be proven either way? –  Koeng Nov 22 '12 at 15:59
@CharlesStewart, I shouldn't have used the word attributes, I meant properties. Wrong or right, I think my answer is clear. The issue with the paradox is the definition of R, you cannot resolve it by saying R is a set for which neither of the two statements 'is a member of R' and not-'is a member of R' hold. Perhaps you should explain what you mean by attributes. –  Cantor Nov 22 '12 at 17:03
@Koeng, provability is another issue. It certainly cannot be both true and false (by the law of excluded middle). –  Cantor Nov 22 '12 at 17:11
But it seems to me that we are not talking about the excluded middle. I think it's different to say that something is "both true and false" and "neither true nor false". Or isn't it? –  Koeng Nov 22 '12 at 17:37
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