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I've heard of some other paradoxes involving sets (ie, "the set of all sets that do not contain themselves") and I understand how paradoxes arise from them. But this one I do not understand.

Why is "the set of all sets" a paradox? It seems like it would be fine, to me. There is nothing paradoxical about a set containing itself.

Is it something that arises from the "rules of sets" that are involved in more rigorous set theory?

EDIT: I don't really have much of a background with Set Theory, so an answer that deals with common english would be much appreciated =)

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Try a finite example, and try to answer, "What is the sum of its elements cardinality?" –  Jonathan Fischoff Jul 20 '10 at 23:17
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Intuitively, you can put many things in a basket -- but one thing you can never put in that basket is the basket itself. –  littleO 2 days ago
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The more one thinks about it, the more there is in it. The question should give a definition of set, and the answer should be based on the def itself, not on some other tall constructs. –  George Chen 13 hours ago

11 Answers 11

up vote 42 down vote accepted

Let $|S|$ be the cardinality of $S$

$|S| < |2^S|$ can be proven with generalized Cantor's diagonal argument

Proof of the set of all sets doesn't exist:

Let $S$ be the set of all sets,

$|S| < |2^S|$

but $2^S$ is a subset of $S$, because every set in $2^S$ is in $S$. Therefore

$|2^S| \leq |S|$

A contradiction. Therefore the set of all sets doesn't exist.

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This is also a manifestation of godel's incompleteness theorem :) –  workmad3 Jul 20 '10 at 23:02
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What does 2^S mean? –  Casebash Jul 21 '10 at 0:51
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@Casebash: 2^S is standard notation for the power set of S: That is, the set of all subsets of S. If S were {1, 2}, 2^S would be {{}, {1}, {2}, {1, 2}}. –  Larry Wang Jul 21 '10 at 0:58
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@Seamus Here S is defined as the set of all sets. Every element of 2^S is a set, implies they are also elements of S. –  Chao Xu Jul 21 '10 at 12:22
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How is that incorrect. We always consider the standard framework when we are not presented other ones. –  Chao Xu Mar 6 '11 at 7:55

As others have explained, the existence of universal set is incompatible with the Zermelo–Fraenkel axioms of set theory. However, there are alternative set theories which admit a universal set. One such theory is Quine's New Foundations.

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Possibly also useful: Elementary Set Theory With a Universal Set, by M. Randall Holmes. –  MJD Mar 18 '12 at 18:12

The "set of all sets" is not so much a paradox in itself as something that inevitably leads to a contradiction, namely the well-known (and referenced in the question) Russell's paradox.

Given any set and a predicate applying to sets, the set of all things satisfying the predicate should be a subset of the original set. If the "set of all sets" were to exist, because self-containment and non-self-containment are valid predicates, the set of all sets not containing themselves would have to exist as a set in order for our set theory to be consistent. But this "set of all sets" cannot exist in a consistent set theory because of the Russel paradox.

So the non-existence of the "set of all sets" is a consequence of the fact that presuming it's existence would lead to the contradiction described by Russel's paradox.

This is in fact the origin of Russel's paradox.

In his work "The Basic Laws of Arithmetic", Gottlob Frege had taken as a postulate the existence of this "set of all sets". In a letter to Frege, Bertrand Russell essentially blew away the basis of Frege's entire work by describing the paradox and proving that this postulate could not be a part of a consistent set theory.

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This is how I had understood it before and I always found it to be the simplest answer, which is quite neat. –  Justin L. Jul 21 '10 at 5:35
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If the "set of all sets" were to exist, because self-containment and non-self-containment are valid predicates, the set of all sets not containing themselves would have to exist as a set in order for our set theory to be consistent - Not true, as Quine's New Foundations, cited by François, shows. It has a universal set, but not the Russell set. –  Charles Stewart Jul 21 '10 at 11:05
    
I guess this depends on what one considers a "set theory". The usual image of the universe of sets, given by the cumulative hierarchy, doesn't allow for the set of all sets. –  Michael Greinecker Mar 8 '12 at 9:13
    
I don't see how you could not consider Quine's New Foundations a set theory; it has the usual universe of sets in it. It may in fact have a superset (superclass?) of ZF's sets, though it's possible stratified comprehension excludes some sets ZF includes. It is akin to NBG set theory; some definite weirdness if you're used to ZFC, but basically the same thing for practical purposes. –  prosfilaes Oct 17 '12 at 6:48
    
NF doesn't necessarily have all of ZF's sets in it. While it's consistent that the von Neumann version of $\omega$ exists in NF, I believe it's not provable that it does. –  Malice Vidrine Jun 23 at 10:12

Russel's paradox arises if you consider the set $U=\left\{x:x\not\in x\right\}$. Ask yourself if $U\in U$. If you suppose so, then by the definition of unrestricted set comprehension $U\not\in U$. You have a contradiction, so it must be the opposite of what you supposed, that is, $U\not\in U$. But this is the same as saying $U$ belongs to the complement of itself, that is, $U\in U$. You now have another contradiction, but this is far worse, since you have no hypotheses. The whole theory is logically inconsistent.

In set theory there are two ways for getting rid of the Russel's paradox: either you disallow the set of all sets and other similar sets (see for example the Zermelo-Fraenkel set theory), or you allow them, but you also restrict the way they are used (see for example the Morse-Kelley set theory).

In the first case, set comprehension says if you have a set $A$ you can have $\left\{x\in A:\phi\left(x\right)\right\}$ (notice: writing $\left\{x:\phi\left(x\right)\right\}$ is just wrong in this case, because you have to have an initial set). If you now define $U=\left\{x\in A:x\not\in x\right\}$ and you repeat the same passages as before, it only follows that $U\not\in A$. There's no contradiction and the theory is consistent.

In the second case, you consider classes, not just sets. Sets are classes that belong to some other class, while proper classes are classes that belong to no class. Set comprehension, in this case, says you can have $\left\{x:\phi\left(x\right)\right\}$, but all its members are sets by definition. If try to reproduce Russel's paradox, you get that $U\not\in U$. If you then suppose that $U$ is a set, then you have a contradiction, so $U$ must be a proper class. This is all you get. No contradictions. The theory is consistent.

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There are other ways to avoid Russell's paradox, too. For example, the axiom schema of comprehension in NF doesn't allow you to construct $\{x: x \notin x\}$ (or, AIUI, even $\{x \in A: x \notin x\}$ in general) because $x \notin x$ is not a stratified formula. It does let you construct the universal set $\{x: x=x\}$, though. –  Ilmari Karonen Mar 8 '12 at 8:57

Perhaps the following image will help take from here:

enter image description here

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Just by itself the notion of a universal set is not paradoxical.

It becomes paradoxical when you add the assumption that whenever $\varphi(x)$ is a formula, and $A$ is a preexisting set, then $\{x\in A\mid \varphi(x)\}$ is a set as well.

This is known as bounded comprehension, or separation. The full notion of comprehension was shown to be inconsistent by Russell's paradox. But this version is not so strikingly paradoxical. It is part of many of the modern axiomatizations of set theory, which have yet to be shown inconsistent.

We can show that assuming separation holds, the proof of the Russell paradox really translates to the following thing: If $A$ is a set, then there is a subset of $A$ which is not an element of $A$.

In the presence of a universal set this leads to an outright contradiction, because this subset should be an element of the set of all sets, but it cannot be.

But we may choose to restrict the formulas which can be used in this schema of axioms. Namely, we can say "not every formula should define a subset!", and that's okay. Quine defined a set theory called New Foundations, in which we limit these formulas in a way which allows a universal set to exist. Making it consistent to have the set of all sets, if we agree to restrict other parts of our set theory.

The problem is that the restrictions given by Quine are much harder to work with naively and intuitively. So we prefer to keep the full bounded comprehension schema, in which case the set of all set cannot exist for the reasons above.

While we are at it, perhaps it should be mentioned that the Cantor paradox, the fact that the power set of a universal set must be strictly larger, also fails in Quine's New Foundation for the same reasons. The proof of Cantor's theorem that the power set is strictly larger simply does not go through without using "forbidden" formulas in the process.

Not to mention that the Cantor paradox fails if we do not assume the power set axiom, namely it might be that not all sets have a power set. So if the universal set does not have a power set, there is no problem in terms of cardinality.

But again, we are taught from the start that these properties should hold for sets, and therefore they seem very natural to us. So the notion of a universal set is paradoxical for us, for that very reason. We are educated with a bias against universal sets. If you were taught that not all sets should have a power set, or that not all sub-collections of a set which are defined by a formula are sets themselves, then neither solution would be problematic. And maybe even you'd find it strange to think of a set theory without a universal set!

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Thanks for the explanation. –  George Chen 2 days ago

An informal explanation is Russel's Paradox. The wiki page is informative, here's the relevant quote:

Let us call a set "abnormal" if it is a member of itself, and "normal" otherwise. For example, take the set of all squares. That set is not itself a square, and therefore is not a member of the set of all squares. So it is "normal". On the other hand, if we take the complementary set that contains all non-squares, that set is itself not a square and so should be one of its own members. It is "abnormal".

Now we consider the set of all normal sets, R. Attempting to determine whether R is normal or abnormal is impossible: If R were a normal set, it would be contained in the set of normal sets (itself), and therefore be abnormal; and if it were abnormal, it would not be contained in the set of normal sets (itself), and therefore be normal. This leads to the conclusion that R is both normal and abnormal: Russell's paradox.

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I mentioned this paradox in my question; it however is not the same one that I am asking about. –  Justin L. Jul 20 '10 at 23:17
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You're right, sorry. Probably you can say that, since "the set of all sets" is not so well-defined, it must necessarily contain the set from Russel's paradox. Since the set from Russel's Paradox can't exist, neither can any set that contains it. But I feel this is cheating somehow. –  Edan Maor Jul 20 '10 at 23:37
    
That's not cheating, that's exactly correct. I always use Russel's Paradox for the "set of all sets" question. The main point in both cases anyway is remembering that Cantor's definition of a set is a little to vague if you dig deep enough. –  balpha Jul 21 '10 at 5:10

How about a set that contains everything with the exception of itself?

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You can still make the cardinality argument that the set cannot contain all its subsets. –  Michael Greinecker Mar 8 '12 at 9:11
    
Insightful! @MichaelGreinecker. In type theory, the set of all the subsets resides on upstairs. –  George Chen Aug 22 at 23:50
    
A set cannot have any of its subset as members because a subset of a set involves the totality also, and therefore must not be a member of the larger set; otherwise there will be a vicious circle. –  George Chen Aug 23 at 0:42
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@GeorgeChen There is something in set theory called a transitive set, and for such a set, every element is a subset. Here is an example: $\{\emptyset\}$. –  Michael Greinecker Aug 23 at 7:25
    
This is wild. In type theory, the subclass stays on the same floor; the member goes to downstairs. They are totally different things. –  George Chen Aug 23 at 9:02

"Why is "the set of all sets" a paradox? It seems like it would be fine, to me."

The lesson of this and other set theory paradoxes is that when you define a set in a proof, you are obliged to demonstrate that the set exists and is well-defined. To me, a paradox of Russell's Paradox is that he fell into this trap since he pretty much wore himself out trying to generate a rigorous foundation for Mathematics.

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The set of all sets is well-defined in the set theory Cantor used: to any logical predicate there is a set of all objects satisfying it. The problem is that Cantor's set theory is inconsistent. –  Hurkyl Mar 8 '12 at 9:36

This turns out to be the same as @donroby's answer. The scheme is inspired by the proof of Cantor's theorem, the use of which is implicit in the accepted answer.

Definition: A set is all the objects satisfying some propositional function.

Let $S$ be the set of all sets, then $S=\{\alpha \ | \ \alpha$ is a set.$ \}$

Let $T = \{ \alpha \in S : \alpha \notin \alpha \}$,

$T$ is a set $\Rightarrow$ $T \in S$

$T \in T \Rightarrow T \notin T$ --- (1)

$T \notin T \Rightarrow T \in T$ --- (2)

(1).(2) $\Rightarrow$ Paradox.

Note: This answer dispenses with such higher notions as "power set," "cardinality" and "less than" whose properties cannot be ascertained before the properties of set are ascertained.

Further analysis leads to type theory which, to my current understanding, is not a rule, but a consequence of the definition.

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PM is not so much about foundations as about a good habit of the mind. The accepted answer went up to the 5th floor to smooth out something in the basement. –  George Chen 2 days ago

Herbert Enderton at the Encyclopedia Britannica writes

Russell's paradox, statement in set theory, devised by the English mathematician-philosopher Bertrand Russell , that demonstrated a flaw in earlier efforts to axiomatize the subject.

Russell found the paradox in 1901 and communicated it in a letter to the German mathematician-logician Gottlob Frege in 1902. Russell's letter demonstrated an inconsistency in Frege's axiomatic system of set theory by deriving a paradox within it. (The German mathematician Ernest Zermelo had found the same paradox independently; since it could not be produced in his own axiomatic system of set theory, he did not publish the paradox.)

Frege had constructed a logical system employing an unrestricted comprehension principle. The comprehension principle is the statement that, given any condition expressible by a formula ϕ(x), it is possible to form the set of all sets x meeting that condition, denoted {x | ϕ(x)}. For example, the set of all sets—the universal set—would be {x | x = x}.

It was noticed in the early days of set theory, however, that a completely unrestricted comprehension principle led to serious difficulties. In particular, Russell observed that it allowed the formation of {x | x ∉ x}, the set of all non-self-membered sets, by taking ϕ(x) to be the formula x ∉ x. Is this set—call it R—a member of itself? If it is a member of itself, then it must meet the condition of its not being a member of itself. But if it is not a member of itself, then it precisely meets the condition of being a member of itself. This impossible situation is called Russell's paradox.

The significance of Russell's paradox is that it demonstrates in a simple and convincing way that one cannot both hold that there is meaningful totality of all sets and also allow an unfettered comprehension principle to construct sets that must then belong to that totality. (Russell spoke of this situation as a “vicious circle.”)

Set theory avoids this paradox by imposing restrictions on the comprehension principle. The standard Zermelo-Fraenkel axiomatization does not allow comprehension to form a set larger than previously constructed sets. (The role of constructing larger sets is given to the power-set operation.) This leads to a situation where there is no universal set—an acceptable set must not be as large as the universe of all sets.

A very different way of avoiding Russell's paradox was proposed in 1937 by the American logician Willard Van Orman Quine. In his paper “New Foundations for Mathematical Logic,” the comprehension principle allows formation of {x | ϕ(x)} only for formulas ϕ(x) that can be written in a certain form that excludes the “vicious circle” leading to the paradox. In this approach, there is a universal set.

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Your answer as originally written was blatant plagiarism. I have improved it so that now it simply does not answer the question and is a bit condescending. You should strive for higher goals. –  Eric Stucky Jun 23 at 10:04

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