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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.

  • 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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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:

  • Okay, so I read everything after "Good news...!" in Professor Farnsworth's voice. – Joseph Weissman Nov 23 '12 at 14:35
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    … 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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The problem lies in the Comprehension principle;

the "recipe" we follow for constructing a set:

M = { x : Fx }

Defining M to be the set of all objects x such that x has the property F.

Bertrand Russell wrote this recipe: R = {x : x ∉ x}

And a paradox arises if you ask if R is an element in R or not!

Therefore Cantor's Set Theory (and Frege's logical system) was proven inconsistent!

Invented by Cantor, the recipe needs adjustment and here's a suggestion.

M = {x : x ∈ M IFF Fx}

Defining M to be the set of all objects x such that x is an element in M IFF x has the property F.

Now x cannot take R as a value in the formula:

R = {x : Defining R to be the set of all objects x such that x is an element in R IFF x is not an element in x. x ∈ R IFF x ∉ x}

And Cantor's Set Theory and Frege's logic are no longer proven to be inconsistent.

You can read more about the Original Set Theory here: https://en.wikipedia.org/wiki/Naive_set_theory#Cantor's_theory

Answer to a comment by jobermark:

Its not "{x: Fx}" that defines "M". The original set constructor does not use the name of the set to be constructed. The naming takes place outside the constructor by use of the identity statement "M={x: Fx}". Enabling inconsistency when F(M) = "M ∉ M".

By using "M" as a metavariable appearing in both sides of the identity statement: M = {x : x ∈ M IFF Fx}, the question whether M is an element of M or not is taken care of INSIDE the set constructor by the contradiction "M ∈ M IFF M ∉ M " and we can no longer make the outside claim that M MUST be an element of M if M is not an element in M.

Not familiar with Metavariables? See the definition of truth:"x" is true IFF x.

It seems Cantor was there before me! In a letter to Dedekind he discussed adjusting the Comprehension Principle ... Thomas Benjamin gives a modern version of Cantors adjusted theory:

Naive Set Theory:

Extensionality: Given two sets A and B , A=B iff (x)[x 'is a member of' A iff x is a member of' B]

Comprehension: Given any predicate P(x), the set {x|P(x)} exists and

(a)[a 'is a member of' {x|P(x)} iff P(a)]

And then Thomas asks if one can still derive paradoxes from this amended version of Cantors Naive Set Theory?

I asked only if Russell's Paradox can be derived!

The predicate is x ∉ x

The name of the set is {x|x ∉ x}

And the set constructor is: x ∈ {x|x ∉ x} IFF x ∉ x

And Cantor then would finally get that for no x is it true that x = {x|x ∉ x}!

But it is confusing to use "{x|x ∉ x}" as the name of the set to be constructed!

So when the predicate is defined as x ∉ x then I stipulate that R = {x|x ∉ x}

So I can have the set constructor with R inside {x|x ∈ R IFF x ∉ x}

And finally R = {x|x ∈ R IFF x ∉ x} (My version)

But we can eliminate R if we want Cantors version:

{x|x ∉ x} = {x| x ∈ {x|x ∉ x} IFF x ∉ x}

  • I like your answer but a reference giving more information on the "Comprehension principle" would help the reader and strengthen your answer. A reference such as plato.stanford.edu/entries/russell-paradox may be beneficial. – Frank Hubeny May 19 '18 at 16:34
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    I did an edit to improve the essay after your suggestions. – Sigurd Vojnov May 20 '18 at 0:31
  • Very good. I edited to add in apostrophes and combined two lines since they were the same sentence. You can roll these back if you want. – Frank Hubeny May 20 '18 at 0:51
  • NO! Im happy to get editor help! – Sigurd Vojnov May 20 '18 at 1:19
  • OK, but M = {x: Fx} alreadly defines M as the set such that x is in M if and only if Fx is satisfied. Your notation combines two equivalent things in a way that makes no sense. Where you have used it, M is not yet defined. It cannot be used to establish its own definition by comprehension. The set of dogs is obviously the set of things that are a dog and have four legs (ignoring the odd mishap), but that fails to define the set of dogs, which is clearly not the set of all things that have four legs. – user9166 May 20 '18 at 7:11
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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.)

  • 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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Obviously, this works, but you need something resembling a reason:

  • You can throw out the Law of the Excluded Middle, resulting in a more basic "intuitionistic" logic.
  • you can insist upon an inherent ordering in the process of creating sets, leading to the elaborate and unnatural construct of ZFC and its relatives
  • you can claim the notion of 'being an element' simply does not apply to certain kinds of objects, including collections that are 'too big' an those that are not "well-founded"
  • you can imply that all references in a language have a natural order of resolution, starting with the approach suggested in the question and leading on to Russel's and Quine's elaborate and unnatural notions of 'ramification' to handle more and more special cases.
  • you can claim negation is not total and admit to some contradictions into your logic as special cases
  • you can go through a massive project of describing all forms of ordering, and seek out a maximal stable model, accepting that containment, being the most basic kind of ordering must be isomorphic to that maximal stable model if it proves unique

All of these, and others, work. And all of them are very hard to accept without a better philosophical basis than the need to solve this issue.

So you have traded no answer for too many answers, and since all of them feel silly, at first blush, you are in no better stead. It remains true that logic in the thoroughgoing classical sense cannot be right about our notions of containment, universality, and negation all as we naively understand them -- in a fully resolved and unchanging Platonic realm. Some part of our natural model is not naturally correct.

The paradox is then what part of your naive and natural understanding you want to discard in order to feel safe, and how we can achieve agreement on something so basic, so that the evolution of math can move forward. And it is no easier to solve than the original problem.

The modern 'solution' has been to bring up all of these solutions and impose one of them by force of numbers -- a bizarre political process alien to the notion of mathematics itself.

0

Bertrand Russell had discovered an inconsistency Frege's yet-to-be-published axiomatization of set theory. Using Frege's axioms, Russell demonstrated that the set of all sets that are not elements of themselves could be both proven to exist and not to exist. Subsequent axiomatizations of set theory have managed to avoid this problem.

Note that the problem is not one of self-reference. It can be trivially shown that, for any binary relation R, there does not exist x such that, for all y, yRx iff not yRy.

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