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## Applied Version: Chinese Lao-tzu's Tao-te-ching

The 1st statement of Lao-tzu's book says "道可道 非常道 A describable statement doesn't hold true for all the time." a statement which is described but if not hold true all the time, it defies what it says.

Sounds a bit closer to the Liar paradox, which is related to Russell but not exactly the same thing (because it doesn't deal with sets). --Trovatore 05:47, 4 November 2005 (UTC)

The English translation is completely wrong, and the quote is incomplete, therefore FAILURE.

Perfect application of asshole chinese culture, wow. voidnature 13:06, 14 May 2011 (UTC)

## how is this a paradox and not just being retarded?

Is it a paradox to say the following: let x = x for all values of x that are not equal to x?

It's not so much a "paradox" as it is simply a very poor definition of terms. It's like dividing by zero. It's as though you were to say, "The definition of 1 shall be that which cannot define 1.", or "The set which contains the things which it does not contain.", or more succinctly, "The set which cannot exist". It's not that it is 'paradoxical' in any way, it's just that the definition of the set itself is flawed, and so subsequently trying to fulfill the requirements of that set breaks down. You can do it in any number of ways that don't look like "paradoxes" per se, that look like gibberish, but still produce the same results:

"The set that contains 1 and does not contain 1." "The non empty set that contains no values." "The set that is not a set."

Wow, fun... But how is that in any way valuable?

"x = x for all values of x that are not equal to x" merely implies that no solutions for x exist. It is no way inconsistent with logic or intuition so it is not a paradox. You might as well say that "x2+1=0 such that x is real" is a paradox too

I don't think it is a paradox. I just don't know what it is supposed to mean. I wouldn't know either what the meaning would be of "let x = x for all values of x such that x = x". It is a meaningless string of words, just like "let 1 = 2 for all values of 3 for which 4 = 5".  --Lambiam 01:05, 28 July 2007 (UTC)
Read this quotation from the article: "The whole point of Russell's paradox is that the answer "such a set does not exist" means the definition of the notion of set within a given theory is unsatisfactory." The point of the paradox is to demonstrate that any theory in which it is possible to define a "set which contain all sets which do not contain themselves" is problematic. The anonymous objector is correct that it is possible to create many trivial paradoxes which appear close to Russell's but they do not demonstrate anything, whereas Russell's demonstrates that Frege's set theory is problematic. Lambiam's objection doesn't really make sense. Russell's paradox is about what is contained in a _well-defined_ set, not about how we define a symbol x. Let S contain all odd integers is a valid, non-empty set. Let S contain all odd integers which are even is a well-defined, empty set. If "Let S contain all sets which do not contain themselves" is a well defined set within a theory, then that theory has a problem. Flies 1 (talk) 18:30, 19 August 2010 (UTC)
I completely second the "just being retarded" comment! When somebody holds ground beliefs that are contradictory we tend to call them retarded. A more subtle (politically correct) mathematical term would be "not sound". Soundness is a property that says that no contradiction can be derived from axioms. What is a definition? a definition is an implicit axiom. Hence it is absolutely no wonder that a new axiom (the definition) can be construed that conflict with other axioms that are held to be true. Amazement at "I cant define what I want?!?!" is merely the realization that definitions are axioms as well. Given a set of axioms, and a definition one could check if the definition preserves soundness of the original system. Definitions that dont (like Russels paradox) are merely implicit axioms that result in a system without soundness. without soundness... i.e. just being retarded... For every definition we make we should check if the added axiom preserves soundness, and if it doesnt, then we should not be amazed that making deductions in a (implicitly made unsound) system are no longer sound (i.e. R is an element of R, and R is not an element of R).... — Preceding unsigned comment added by 213.49.89.185 (talk) 00:32, 20 May 2012 (UTC)
As an example of a definition and the sentence of the corresponding axiom: Suppose I define the proposition/relation
• OnUnitCircle(x,y):=(x^2+y^2=1) for real x,y
Then this is completely equivalent to
• (∀x,y∈R)(OnUnitCircle(x,y)<=>(x^2+y^2=1))
Note the conversion from ":=" to quantifiers.... — Preceding unsigned comment added by 213.49.89.185 (talk) 00:51, 20 May 2012 (UTC)
From this perspective set theory is probably not the only place where definitions cause formal systems to lose soundness, hence it would seem to be more usefull for logicians to refer readers to algorithms to check a formal system of axioms/seduction rules for soundness when adding a specific definition/axiom and point out the conflicting definitions and axioms. Or to search or design such algorithms when they dont exist yet. — Preceding unsigned comment added by 213.49.89.185 (talk) 05:03, 20 May 2012 (UTC)
A very large amount of work was done in the early decades of the twentieth century to try to produce just such an algorithmic approach to proving consistency of axiomatic systems, following David Hilbert's approach. Alas, it was eventually proved that no such algorithm can cover an axiomatic system capable of providing the necessary support for analysis. JamesBWatson (talk) 09:28, 11 May 2013 (UTC)

Nice this discussion.. I think the following argument has not been stated so far:

If you say "x := 3+2" this is a definition.

If you say "x = x*x - 6" this is an equation that may have any number of solutions (in this case, it has exactly one). Some equations have no solution, or an infinite number of them. In a way we can say a definition is an equation with one unique solution.

To define a set, you would say "M = {x | A(x) }" where A is a statement. Normally, this is a definition.

However, if the statement A refers to something on the left side of the "=", the whole thing becomes an equation where M is the variable, and therefore it doesn't need to have a solution any more. For the Russel set "definition", the reference to the object M exists, though it may not be visible at the first sight. So the "definition" is simply an equation that has no solution. Fine?

yes, this is the truth, thanks for stating it so clearly :)

The problem of the paradox is (I think) that its mostly about the consistency of the language to describe sets. In English, we're always speaking in terms which by some view might seem inconsistent, but we've got a common culture and set of ideas that we can correct for any inconsistencies so most ideas can, in general, be communicated. When we speak in mathematics, we speak of absolutes. In a language of absolutes there can be no inconsistencies. However, using the theories of sets at the time, Russell described such a set. Once a flaw like that has been established to exist, one has to wonder about mathematics as a whole. Two plus two will probably still equal four, but some of the more advanced theories you or someone else may have discovered may in fact now be proven false by this new discovery (because their "truth" may rely on the inconsistency).
Don't get from this article that there's something wrong with defining a entity in terms of itself. Think of the perfect numbers, where we discuss the set {x is an Integer | x = \sum n, n is an integer, n divides x, 1<=n<x}. Without this ability we'd be very limited in power. You just need to prove a solution exists to make sure you are discussing a topic of any worth. In this case I can give at least 6 = 1 + 2 + 3 and 28 = 1 2 + 4 + 7 + 14. Otherwise, you might be discussing the null set, from which you can prove has any property you desire! Why? There's nothing in the set to prove otherwise.
Btw, you never said what "type" x is in your original argument. If x is a natural number, you're fine. If x is real or complex, x^2 -x - 6 either has two real distinct real solutions or it has none since x^ -x - 6 is not a quadradic. It factors as (x+2)(x-3). For quaternions, there's infinitely many. For any other type we'd probably have to define what "*", "-", and "6" mean for that type. Root4(one) 23:55, 20 January 2007 (UTC)

## Principles of Mathematics vs. Principia Mathematica

1903 for the former, 1910 for the latter: Principia Mathematica is really "later". Randall Holmes 06:04, 29 December 2005 (UTC)

## Meta-Sets

A set which contains sets is not a set, it is a meta-set. When you define a set you stand above all the elements in existence and you group some and exclude others, and the set being defined is on a higher level than those elements and cannot be included in itself. When you define a set you take a virtual lasso which surrounds the defined elements, and a set cannot be a member of itself just as a lasso cannot surround itself. The preceding unsigned comment was added by 60.234.145.146 (talk • contribs) 13:13, 1 March 2006.

Well, right, sort of. You're very close to the intution that underlies contemporary set theory; you're just using nonstandard language.
Going with your language for now: The point is that "meta-sets" are very important; we need them, whatever we call them. And meta-meta-sets too. And so on (a very long "and so on"; we iterate through all the finite and transfinite ordinals).
Now, a couple of things: First, we find that we don't have to start with any actual objects, and it simplifies our life a bit if we omit them. We still get one set, which is the empty set, two meta-sets, four meta-meta-sets, sixteen meta-meta-meta-sets, 65536 meta4-sets, and so on.
Finally, we want to consider all meta-to-the-whatever sets all together; the convenient thing to call them is just simply "sets", and henceforth we drop the "meta"s.
We've just reinvented the von Neumann universe, into which all known mathematics can be coded. --Trovatore 15:15, 1 March 2006 (UTC)
Why is a set of sets not a set? You could have, say, a set of tennis rackets. And you could have multiples of those sets, each from a different tennis racket manufacturer. So what do you call all of those sets? Well, a set of sets. In no way a "meta" set. Just a set. Because you could refer to those sets of tennis rackets by their manufacturer's line, and not even call ti a "set". Likewise, you could call a tennis racket itself a "set of materials and geometric shapes". Now you have sets of sets of sets, but none of them are in any way unreal. A set is a set. And the set of sets is a set. "The set of all sets that does not include itself", however, is simply an improperly defined set. Or maybe an eigenset? —The preceding unsigned comment was added by 66.159.227.60 (talk) 02:07, 2 April 2007 (UTC).

One has to be careful when abbreviating a term such that it is the same as a different term. To avoid confusion one should only abbreviate "metan-set" to "set" when it is clear from the context that it has been abbreviated. That is; M={A|A is not an element of A} should be transcribed as "The meta-set of all sets which do not contain themselves", so that it is clear that while M does not contain itself, this does not mean that it in fact does contain itself (by function of the set definition) since it is not a set, it is a meta-set. The preceding unsigned comment was added by 60.234.145.146 (talk • contribs) 16:56, 2 March 2006.

It betrays a certain lack of humility to attempt to dictate linguistic usage to the entire mathematical world. What I'm explaining to you is the accepted terminology.
Your substantive point, on the other hand, is basically correct. The Russell paradox does not show up at all in the conception of set embodied in the von Neumann universe. That's one of the big reasons that this conception has won out over Frege's. --Trovatore 18:00, 2 March 2006 (UTC)
A set of sets is a set. Each set in the set is an element. The set of sets is "definable" and thus it is a "set". And I do believe sets can contain themselves. Read the natural numbers article. Now take the construction of natural numbers in set theory where S(a)=a ∪ {a}. Then, lets start with ∅. Then, keep making sure "this set contains itself". You end up with $\aleph_0$. Is $\aleph_0$ consistent? What do you think? voidnature 12:35, 14 May 2011 (UTC)

## Original Research

Isn't most of this article original research? I doubt that anything has been published in a respectable journal which compares Russell's sets to Wikipedia articles. Ken Arromdee 16:13, 6 March 2006 (UTC)

• That's one section.
• The purpose of an enyclopedia is to explain; examples are useful. WP:OR is intended to stop novel interpretations; unless the analogy is wrong, this is not novel, merely good writing.
• It is self-reference, which is also deprecated; it should be possible to mirror this article anywhere without reference to WP as such. I have made a stab at fixing this. Septentrionalis 17:20, 6 March 2006 (UTC)
I was just looking through the 'no original research' policy (WP:OR), and it has a very broad definition of 'original research'.
Regardless of whether the article violates the "No original research" policy, I'm fairly sure it violates the Citation/Verifiability policy. The only place that I could find any references was the history section. The whole article needs to be fully referenced or it violates the policy. For example, the "Independance from excluded middle" section (which I believe is incorrect, see below) doesn't give any information on where that opinion/argument can be found in a reputable publication.
I should mention here that I am new to Wiki editing, so I apologise if I have made a mistake or broken etiquette in some way. Could a more experienced editor please take a look and venture an opinion about this? If there are others who think that this article content is questionable, perhaps a dispute template should be included at the top of the article. DonkeyKong the mathematician (in training) 05:12, 2 May 2006 (UTC)

## Remove: Responses illustrated

remote it Full Decent 04:03, 28 March 2006 (UTC)

## Barber

The barber is a woman.

From the article: "ZFC does not assume that, for every property, there is a set of all things satisfying that property. Rather, it assumes that for any given set and any definable property, there is a subset of all elements of the given set satisfying the property. The object M discussed above cannot be constructed like that and is therefore not a set in this theory".

Excuse my ignorance, but could somebody describe to a non-mathematician how exactly this addition of a "given set" removes the paradox and why M cannot be constructed this way? It's not that obvious to me... let's take a trivial case, and say that this "given set" is M itself, and the property is $P(A) = A \not\in A$. The subset of M satisfying the property P(A) is again M, i.e. the paradoxical set itself.

The phrasing of the excerpt above doesn't seem to preclude choosing M as the "given set". In fact the only reason I came to this article was hoping to find out how formal set theories eliminate this paradox, yet it was only briefly mentioned and I wasn't able to fully understand it from the short description above. --Grnch 17:23, 20 April 2006 (UTC)

The ZFC version of the axiom of comprehension (axiom of separation, axiom of subsets, Aussonderung; these are just different names), says "for every set x and every formula φ, there is a set y consisting of exactly the elements z of x such that φ(z) holds". So to use this axiom to prove the existence of a y having the properties you want, you first have to prove the existence of the relevant x. So if you try to use this to prove the existence of the Russell set, you simply can't get started; you have to prove M exists before you can prove M exists. The discussion I've given you is slightly imprecise but it's the basic idea; hope it helps. --Trovatore 17:34, 20 April 2006 (UTC)
It does help, thanks. I guess I took the meaning of "for any given set" a bit too broadly (since the domain of allowable values wasn't explicitely stated), whereas it only applies to sets definable in ZFC. I appreciate the quick response. --Grnch 18:06, 20 April 2006 (UTC)
Well, actually it applies to all sets, definable or not. What I was arguing is that the proof of the contradiction doesn't go through, when unrestricted comprehension is replaced by separation.
I think you may need to get a clearer understanding of distinctions like syntax/semantics, provability/truth, definability/existence. By the way your original question talks about "how formal set theories eliminate this paradox"; I think that's a misimpression. It's not the formality per se that resolves the paradox, but rather a different conception of set, the one from which the von Neumann universe arises. --Trovatore 18:19, 20 April 2006 (UTC)

## Falicy in "Independence from excluded middle"?

I think that the author of the "Independance from excluded middle" passage has made a mistake.

...

Often, as is done above, showing the absurdity of such a proposition is based upon the law of excluded middle, by showing that absurdity follows from assuming P true and from assuming it false. Thus, it may be tempting to think that the paradox is avoidable by avoiding the law of excluded middle, as with Intuitionistic logic.

On the contrary, assume P iff not P. Then P implies not P. Hence not P. And hence, again using our assumption in the opposite direction, we infer P. So we have inferred both P and its negation from our assumption, with no use of excluded middle.

...

This argument appears to make use of the same reasoning as the earlier examples. At first I thought that none of the earlier arguments really used the law of excluded middle, but then I realized that this argument (quoted) implicitly uses the law of excluded middle.

This argument goes: "...assume P iff not P. Then P implies not P..." The reasoning is fine so far...

"... Hense not P..." This is where the reasoning is flawed, since we have not assumed P, so the fact that P => ~P does not assist us, and thus we cannot conclude ~P.

In other words, since we are not assuming the law of excluded middle, it is possible to assume that neither P nor ~P are true (is it correct to talk about truth in this context?).

Thus the fact that we can show both P and ~P to be self contradictory does not lead to a paradox, since without the law of excluded middle, they can both be false at the same time. DonkeyKong the mathematician 04:05, 2 May 2006 (UTC)

No, P and ¬P can't exactly be false at the same time for an intuitionist; it's just that neither has to be true. (P → ¬P)→¬P is valid in intuitionistic logic. Intuitionistically, you should think of A → B as meaning "I have a way of transforming any proof of A into a proof of B", and think of ¬A as meaning "I have a way of transforming any proof of A into a proof of false". Now if you have a way of turning a proof of P into a proof of ¬P, and you also have a proof of P, then you can put them together to get a proof of P∧¬P, which then gives you a proof of false. Thus you can turn a proof of P into a proof of false; that proves ¬P. --Trovatore 05:38, 2 May 2006 (UTC)
Just the same, you have a point about the passage as written. The deduction of ¬P from P ↔ ¬P is valid, but the deduction of P from the same assumption, as far as I can tell, is not; what you actually get by the same argument is ¬¬P. But this does give us a contradiction, namely ¬P∧¬¬P. And of course from that you can deduce P, because ex falso quodlibet is intuitionistically valid. But the paragraph should be corrected. --Trovatore 18:07, 2 May 2006 (UTC)
Okay, I'm totally out of my depth with this stuff, so I'll just have to take your word for it. Anyway, I think we need an expert on the topic to help us out with this. DonkeyKong the mathematician (in training) 01:31, 19 June 2006 (UTC)
The passage as I found it was correct, although it seems to me you can, in fact, deduce P directly rather than using indriect proof a second time to get ¬¬P. I amended it to try to make the key step clear, let me know if it is still unclear (or if I have made it incorrect). 192.75.48.150 14:36, 19 June 2006 (UTC)

## Arithmetic

The article says that arithmetic is "incomplete". I checked Completeness and Arithmetic but found no explanation: in what way is arithmetic incomplete? MrHumperdink 21:22, 7 May 2006 (UTC)

Thanks for catching that: The flat statement "arithmetic is incomplete", as contextualized (or more to the point not contextualized) in the article, is pretty meaningless. The probable intended meaning is that any (computably enumerable) axiomatization of arithmetic must be incomplete; that is, if you write down any finite list of first-order axioms for arithmetic (or more generally, an infinite list capable of being produced, in principle, by a fixed computer program), then there must be a statement of arithmetic that can neither be proved nor disproved from your list of axioms, or else the list is inconsistent (in which case every statement can be both proved and disproved). See Gödel's incompleteness theorems. Be warned that the article is in pretty bad shape (though listed, inexplicably, as a "good article"). --Trovatore 21:35, 7 May 2006 (UTC)
Thanks for your quick response! So, in layman's terms (if that's possible - calculus is clear as water compared to some of this stuff), arithmetic, by definition, has some property that can neither be proven nor disproven... I understand the idea, I suppose, but what would a "statement of arithmetic" be? MrHumperdink 00:23, 8 May 2006 (UTC)
Not exactly a "property that can't be proven or disproven", but rather that for any fixed set of axioms (subject to the technical stipulations I won't repeat), there's some true statement about arithmetic that's not captured by those axioms. A "statement about arithmetic" is a statement that talks about the natural numbers, using the usual operations and relationships (addition, multiplication, less than). So for example the infinitude of primes is a statement of arithmetic; it says: For every natural number n, there is a larger natural number m, such that m is not the product of natural numbers k and p, both bigger than 1. --Trovatore 03:23, 8 May 2006 (UTC)
May I ask how is that paragraph regarding Gödel's incompleteness thorem relevant to the rest of the page? 212.179.75.202 (talk) 05:41, 27 September 2009 (UTC)
Hmm, well, that's a decent question, I guess. It is kind of far afield from the Russell paradox. However, I think we do want to discuss what the logicists did to try to repair the damage the Russell paradox dealt to their program, and at that point you kind of have to mention Goedel to finish off the story. I think. Any other opinions? --Trovatore (talk) 06:42, 27 September 2009 (UTC)

--

It is a fair question. So I go back to Goedel 1930 Some mathemathematical results on completeness and consistency and find that in the very first sentence Goedel invoked Principia Mathematica "with the axiom of reducibility or without ramified theory of types" (Goedel 1930 reprinted cf footnote 1 in van Heijenoort 1967:595) -- none of which would have existed if Russell hadn't been confounded by the paradox in his name.
More to the point, in his 1931 On formally undecideable propositions Goedel invokes the Richard antinomy, the Liar paradox, and again in a footnote adds "Any epistemological antinomy could be used for a similar proof of the existence of undecidable propositions." (Goedel 1931 reprinted cf footnote 14 in van Heijenoort 1967:598). In an attempt to justify the axiom of reducibility Whitehead & Russell's Principia Mathematica has a detailed discussion of these paradoxes (Chapter II section VIII The contradictions), exepting the Grelling (cf Grattan-Guiness 2000:336).
In his 1934 lectures (a refined presentation of his 1931) Goedel, in his 7. Relation of the foregoing arguments to the paradoxes now invokes the Epimenides paradox. He comments "The solution suggested by Whitehead and Russell, that a proposition cannot say something about itself, is too drastic. We saw that we can construct propositions which make statements about themselves, and, in fact, these are arithmetic propostions [etc]... The [Epimenides] paradox can be considered as a proof that "false statement in B: cannot be expressed in B." (Goedel 1934 reprinted in Davis 1965:63-64). In yet another footnote he invokes Tarski 1933, 1944 and Carnap 1934 (footnote p. 23, page 64). So here we see a continuation of the problem of the paradoxes (Kleene 1952 invokes them to the same degree of depth as Whitehead and Russell 40 years before).
So one point is that the paradoxes, of which Russell's was a kind of prototype because it is so simple, were the fundamental tool that Goedel invoked in his 1931 proofs. Another point is that the difficulty of the paradoxes didn't vanish with Goedel in 1931. I just got Grattan-Guiness 2000 The Search for Mathematical Roots, 1870-1940 a few days ago. This is a hugely dense and chewy book of some 600 pages that begins with DeMorgan and Boole, discusses Frege and Cantor and Russell and the paradoxes (the history of the Russell paradox in great depth) and ends with Goedel 1931. I'm going to need some time to assimilate this puppy. Bill Wvbailey (talk) 16:38, 27 September 2009 (UTC)

## keep it simple

M is already defined. If it includes itself,first you have defined a different set, secondly you have a self referencing object which never should have been allowed. Example: M={A,B} the original. M={A,B,M} a different set and one that cannot be resolved/expanded (what's in M?). M={A,B,M}, M={A,B,{A,B,M}}, M={A,B,{A,B,{A,B,M}}}, the infinite process again. An object can't be in two mutually exclusive sets. A thing can't be listed and unlisted. I've seen the references to the qualifiers x is not x,and you wonder where is reason? phyti--jun 05 2006

1. 'An object can't be in two mutually exclusive sets.': What about an empty set?
2. 'A thing can't be listed and unlisted.': i don't know what you mean by 'listed' and 'unlisted'?
3. 'I've seen references to the qualifiers x is not x, and you wonder where is reason?': i don't know what you mean here.
Please excuse me, for I admit that I'm only a learner in mathematics. But, from what I gather from the Wikipedia article, is this the description of Russell's paradox?
$M \in M \Leftrightarrow (M = \{M|M \in M\} \Rightarrow M = \{\emptyset\})$, but $\forall M \exists M_m \subseteq M$ such that $M_m = \{ M \} \ne \{ \emptyset \}$. --Pyenos 00:57, 26 January 2007 (UTC)
At the left: If $M \in M$, then how does that imply M is empty? Pomte 09:02, 27 January 2007 (UTC)
Let M be the set of all sets that do not contain themselves as members(from the Wikipedia article). If M in M, then M is empty(ie. M={emptyset}). But there is some M whose element is M={emptyset}.
What I mean is that for all M, there is M={emptyset} such that M in M(ie. M={M={M={emptyset}}} and so on). I have to think about this. I'm not sure. --Pyenos 12:58, 27 January 2007 (UTC)
The proof would normally go: $M \in M \Rightarrow M \not\in M$. Regardless of whether M is in or not in M, there are still other sets that belong in M, such as A = {1}, B = {1, 2}, C = {M}, D = {emptyset}, etc. So I'm not sure how you are concluding that M is empty. I am also a learner in mathematics so don't take my word for it. Pomte 17:01, 27 January 2007 (UTC)
Sorry about the confusion. It derives from incomplete understanding that I have on set theory. Please ignore all of my previous contribution to this subsection. I've found out that emptyset is not equal to {emptyset}. I will get back to this after I finish at least one book on basic concepts of math. I'm sorry for the confusion, again. --Pyenos 02:33, 28 January 2007 (UTC)
Well, I do believe sets can contain themselves. Read the natural numbers article. Now take the construction of natural numbers in set theory where S(a)=a ∪ {a}. Then, lets start with ∅. Then, keep making sure "this set contains itself". You end up with $\aleph_0$. Is $\aleph_0$ consistent? What do you think? voidnature 12:24, 14 May 2011 (UTC)

## An odd creature

Some months ago I created the article List of every Wikipedia list that does not contain itself. This is in the spirit of part of the discussion in this article, the more precise in the self-reference. In any case, it simply redirects here; it might be something like a slight easter egg on Wikipedia, but please leave it be (it may not be a brillant joke, but it's worth a couple tens of bytes in the WP database.

Updating my userpage, an odd little self-reference occurred to me: List of every Wikipedia list that contains more items than this list. I know this is a digression from article discussion, but I'm trying to get a handle on exactly what kind of creature this hypothetical article is. It's not quite a Russell paradox, nor quite a Curry paradox.

Here's the issue, in case it's not immediately obvious: whether or not this list is "a problem" depends on the world external to its definition. If the world is certain ways, it's a perfectly ordinary collection. If the world is other ways, it's a paradox. Let's demonstrate:

• Suppose Wikipedia contains ten lists: five of them list 3 items each; five of them list 20 items each. No problem at all arises here, we just put the five big lists on the new list (bringing Wikipedia to eleven lists total, one of them containing five items).
• On the other hand, suppose Wikipdedia contains ten list: four of them list 3 items each; six of them list 5 items each. Now we have a problem. If we include the 5-items lists, this list has 6 items, and all the 5-item lists must be removed. If we leave off the 5-item lists, this list has zero items, and all the 5-item lists must be added (we might initially add the 3-item lists as well, but taking them off once this grows creates no special problem).

Is there a well-known name for logical/empirical paradoxes of this sort? I.e. ones that are only contingently paradoxical? LotLE×talk 06:32, 20 June 2006 (UTC)

### Possible worlds

Hmm... There are plenty of stable states that this list could take. ie. say thre were 4 lists of 3 items, and 10 lists of 15 items, we'd be fine. To me this seems rather ordinary, "paradoxically" speaking... consider the "list of points at which lines ax+b and cx+d intersect?" depending on the state of a b c and d, we could have many solutions, one solution, or no solution... which seems to be exactly the same sort of outcome as this list you've just proposed. A good mathemetician might be able to generalize it with a formula, or failing that there's always computer assisted exhaustion to fall back on. - Rainwarrior 13:04, 20 June 2006 (UTC)

Certainly, there are many states of the world, or "possible worlds" where the paradox does not arise. In fact, I'm pretty sure that in a measure theoretic sense, the measure of the set of possible worlds where the paradox does arise is zero. Nonetheless, one can easily find an enumerably infinite set of "problem" possible worlds. Let's call worlds that are paradoxical in the described sense "L-paradoxical". The second part of this is almost immediate, by a trivial variation of the prior example:
• Suppose Wikipedia contains ten lists: four of them list 3 items; five of them list 5 items; one of them, BIG, lists "many" items. For every value of N > 5, the possible world described is L-paradoxical. That is, BIG must surely be included in "LoeWltcmittl". But if we try to put all the 5-item lists in LoeWltcmittl, we get a problem. So there's a countable infinity of problems. (and infinitely many other families of "problem worlds" are easy to construct).
The measurement thing is slightly more involved, but not too much. Basically, a possible world (down to homomorphism) is defined by a set of ordered pairs of natural numbers: <NumMembers, SizeOf>. That is, a (homomorphic equivalence class of) world(s) is described by the number of lists of each size that are in it. For example: "4 3-item; 5 5-item; 1 20-item". So described, some worlds are L-paradoxical, and others are not. Notice that the possible worlds are enumerable.
• We can consider all the worlds ranked by total size: worlds of 1 list (or whatever size), worlds of 2 lists, etc. If a world of size M has an "instability point"—that is, a number of items in LoeWltcmittl that would create an L-paradox—that point must be some number ≤ M. If every list in the world contains more items than M, no instability is possible. However, since M is some particular finite number, the measure of M-sized worlds in which no list is as short as M is exactly 1 (natural numbers keep going up, after all, any initial segment is measure 0).
In practical terms, one might be surprised to find billion, or trillion, or googleplex length lists on Wikipedia, but formally there is no size bound. Of course, some practical system, like WP that has an extreme bias towards "small" lists (for any value of small; say, N < number of particles in the universe) is likely to be L-paradoxical with measure greater than zero.
But all of that is not really what I was asking. My original point was that it is interesting that a construct is not just empty, or just undefined in a simple way, in some possible worlds; rather the construct is paradoxical in some possible worlds... but perfectly ordinary in other possible worlds. In some worlds (infinitely many, in fact), LoeWltcmittl cannot contain any particular number of items (including zero), and yet it gives a precise inclusion criterion for any particular possible member.
I'm familiar with paradoxes that are paradoxical by their actual form, i.e. in every possible world. But it is somewhat novel to me to have stumbled on a paradox that is, as I say, contingently paradoxical. Of course, I'm sure someone has thought of this type of thing before... so I was just hoping to learn that this was already known as, e.g. "Jones' Paradox". LotLE×talk 19:14, 20 June 2006 (UTC)
Well, it's an interesting construction to say the least. I've been reading List of paradoxes lately (that's how I ended up here), and so far I don't think anything has been "contingent". Mind you, paradox doesn't always mean "self contradicting", it more often has the sense "unintuitive but true" (ie. the Monty Hall problem is considered a paradox). (How come you created a sub-heading on the talk page for my reply? O_o?) - Rainwarrior 03:39, 21 June 2006 (UTC)
Arguably, particular instances of Curry's paradox are contigently paradoxical when their consequents are contigent. For instance the sentence "If this sentence is true, then Santa Claus exists" would be unproblematic if Santa Claus really did exist (which he does), the sentence is simply true. 192.75.48.150 18:25, 21 June 2006 (UTC)
I thought the "paradox" part of it was that you can prove anything to be true regardless of whether or not it is true... but perhaps our definitions of "paradox" differ. Is a paradox merely some form of contradiction? I thought we call Curry's paradox a paradox not because it is self referential or has a contradiction, but because it is an unusual or unexpected consequence of systems of logic. - Rainwarrior 01:33, 22 June 2006 (UTC)

I thought about 192.75.48.150's comparison some. There is a slightly different pattern to it when the consequent in Curry's paradox is already true. For example:

If <this line> then Haskell Curry exist(ed)


But in the actual world, the consequent is true, so:

If <this line> then TRUE


Which means that <this line> is going to come out TRUE, regardless of the value of <this line>. I.e. 'FALSE → TRUE' and 'TRUE → TRUE'. But in particular, since <this line> is the implication, it turns out true. We don't exactly prove some false (or self-contradictory) consequent; but what we do is follow the same reasoing that is normally the fallacy of Affirmation of the consequent. Just by having a true consequent, we manage to prove the antecedant: that's a slightly different no-no that proving a false consequent, but it's still a no-no.

So the kind of contingency suggested is still quite a bit different than that in the "L-paradox". In that, some possible worlds are entirely clear sailing, while no case of Curry's paradox is problem-free. LotLE×talk 03:20, 22 June 2006 (UTC)

## Fuzzy Russell

I realized that Russell's paradox can be resolved if you use fuzzy sets, then you can just say that the set of all sets that don't contain themselves contains itself halfway.--SurrealWarrior 19:19, 12 July 2006 (UTC)

At first I was going to disagree, but after thinking about it, I think it might work. That's kind of weird... if there's a 50% chance that it contains itself (a, with a value of 1.0) and a fifty percent chance that it doesn't (b, with a value of 0.0), and (a) implies (b) and (b) implies (a) as the problem's definition, resolution of these implications brings it to the same state: 0.5 * 1.0 + 0.5 * 0.0 = 0.5, so any further resolution will again reach the same state, whereas using any other value than 50% will cause it to oscillate. ... Weird. - Rainwarrior 19:28, 12 July 2006 (UTC)
Fuzzy set's do not define a probability, they define a degree of membership. In classical sets the degree of membership is either 1 or it is 0, it is either in a set or it is not. Using fuzzy logic to solve this;If the truth of the fact that the set contains intself is T(S) then following $T(S) = T(\bar{S}) = 1 - T(S) \therefore T(S) = 0.5$. In classical set theory 0.5 is not a valid answer, it must either be true, 1, or false, 0, which is where the contradiction occours. Fuzzy logic however lets us assign a degree of membership which in this case would be 0.5. Refer to Ross TJ 2004, 'Fuzzy Logic with Engineering Applications', 2nd edn, Wiley and Sons, West Sussex; specifically page 134 where this exact problem is discussed and where I got my equation from. K2kingy (talk) 12:40, 17 April 2012 (UTC)
This isn't surprising since any linear feedback system so constructed will either end up "stuck in the center" or oscillate. But "stuck in the center" means you've allowed trivalent logic, i.e. an introduction of an acceptable third state (0.5) to sneak into your theory. cf Kosko 1993 Fuzzy Thinking, Hyperion, NY, ISBN 0-78678-8021-X. Russell's paradox is a paradox of bivalent logic. Maybe this article isn't clear on that. The deeper problem is this: Boolean logic encourages a mixture of the arithmetic with the logical signs, but the logical sign "0.5" does not exist in that theory, nor does arithmetic equality, i.e. you should be using the logical sign IFF (IF & ONLY IF, the biconditional), i.e. ↔ ≡ (P & Q) V (~P & ~Q), where the sign ≡ means "is defined as", and &, V, ~ are the usual logical signs. (Kusko does not make this mistake, see p. 26 where he specifies biconditionality). The way to avoid this blunder is (i) not use Boolean equivalents, and (ii) to set up a formal truth table with the above equation for IFF. Now allow only "1" and "0" or better yet two novel signs such T and F or ■ and □ and keep in mind that the behavior of the signs is determined by the theory (i.e. the truth tables in the logical system of only two values). What will happen is the truth table will fail to produce a tautology (all 1's beneath the ↔). In fact what happens is the table yields all 0's in its two rows. The table is constrained: only two rows of the four are permitted (P ≡ ~Q, ~P ≡ Q: we assign the truth values 1 and 0 per the definition of ~: P=1, Q=0, and P=0, Q=1). Given these constraints the truth table yields 0 for both of these rows, meaning neither row can be satisfied (i.e. always false). BillWvbailey (talk) 16:08, 17 April 2012 (UTC)

## A closer and clearer Russell paradox by Ali nour mohammadi sharif university of tehran

we say that sets A1,A2,...,An is a chain of set A1 if A(i+1) be in A(i) for i=1,2,...,n-1 and chain can be infinite(in this case each Ai has an infinite chain). Let B=set of All sets that have no infinite chain ,so B has no infinite chain. so B is in B ,but in this case B has the infinite chain BBBB... And this is Paradox. --213.217.57.212 17:53, 17 July 2006 (UTC)onourmohammadi@yahoo.com

## Urban Legend

There is an urban legend, at least at my undergraduate school, that several mathematicians killed themselves after hearing about Russell's Paradox. Probably not true, but there aren't many good urban legends about math. Has anyone else heard this?

No, but they must have died only to encounter even more paradoxical incomplete daemons in hell. See the first in the List of unusual deaths. Pomte 17:01, 27 January 2007 (UTC)
Although I was 90% certain the current (As of March 22) No. 2 spot was No. 1 as of January 27th, I had to go to the 27th of January List of unusual deaths page just to be 99.5% certain you were talking about who I thought you were thinking about. At least here we have the history pages to consult.
Root4(one) 07:19, 22 March 2007 (UTC)

## what?

Russell's paradox came to be seen as the main reason why set theory requires a more elaborate axiomatic basis than simply extensionality and unlimited set abstraction. The paradox drove Russell to develop type theory and Ernst Zermelo to develop an axiomatic set theory which evolved into the now-canonical Zermelo–Fraenkel set theory.

not that my english is that bad but....is that english?--Lygophile 09:26, 13 November 2006 (UTC)

Seems OK to me. Any particular trouble with it? EdC 09:42, 17 November 2006 (UTC)
"came to be seen" is a tad awkward, especially in the opening paragraphs. "was a primary motivation for the development of higher-complexity set theories" might be better, though I'm not sure if it's accurate as given. (Certainly Russell's paradox, along with Cantor's and a few others stimulated their development.) 12.103.251.203 05:57, 22 March 2007 (UTC)
Thanks, I've used that fragment. A little clearer now? –EdC 23:30, 23 March 2007 (UTC)

## Sancho Panza as a governor

It seems to me that the Russell's paradox was known much before Russell 'discovered' it!

http://quixote.rincondelvago.com/2_51/

So what we see here has something in common with the Russell paradox, in that it gives a proposition that if you assume it's true, you can immediately derive that it should be false, and vice versa (well, more or less; there are some complications involving the will of the prisoner versus the choice of the authorities that make it not quite pure).
In any case, sure, that idea predates Russell, in many forms. The liar paradox is another similar thing, and a form of that one even made it into the Bible, long before Cervantes.
The novelty in Russell's case was specifically that he applied the idea to refute Fregean set theory. If you don't have that piece of context, there may not seem to be much point to the paradox. --Trovatore 17:22, 5 April 2007 (UTC)

## Accessibility

I have looked at a few maths articles in WP and find that they are rather inaccessible to the mythical "average" reader. I would like to change it a bit so that the first thing was an "everyman" explanation of the paradox itself with an example and then the "Part of the foundation...." bits. e.g. Bertrand Russell described a paradox involving the definition of sets. A set can contain elements, some of which may be sets. For example: Think of all sets that are dog breeds such as L={Labradors}. Think of all these sets grouped as a single set D ={all sets of dog breeds}. Additionally, we could have a set E={all sets that are NOT dog breeds}. The set E is special because it has itself as a member i.e. ONE of the sets that are NOT dog breeds is set E. Now think more generally of M={all those sets like D that do not contain themselves as members}. If M does not contain itself, then M has to be included in the list of sets that do not contain themselves. But if it is included then it WILL contain itself and thus not be listed. This leads to the paradox that the statements "M is an element of M" and "M is not an element of M" seem to be both true and false at the same time. In set notation: etc Diggers2004 05:38, 11 April 2007 (UTC)

Reverted changes: I like to encourage my Year 7 students (and their parents - often with less mathematics) to use Wikipedia. Mathematical specialists often have a different (and presumably correct) conception of the meanings of mathematical terms than does the general public. e.g. the Encarta Dictionary has the following: contradiction in set theory: the contradiction in set theory resulting from assuming that it is possible to form any set whatsoever, contradicted by the set of all and only things that are not members of themselves (Microsoft® Encarta® 2006. © 1993-2005 Microsoft Corporation. All rights reserved.)

Could you suggest a start for the article that explains the crux of the paradox with the minimum reference to pre-requisite knowledge ? One that a "general" reader might understand and then move onto the more complete and complex details ? Diggers2004 07:12, 17 April 2007 (UTC)

I think that's a worthy goal, but we need to keep some things in mind. This is a reference work, not a textbook, and we want to maintain an encyclopedic style. That entails that you start the article with as accessible a summary as possible, but a summary, not an example, not a warm-up exercise.
Beyond that, I think it's important to emphasize that the reason the Russell paradox specifically (as opposed to, say, the closely related liar paradox) is important, and that comes from the historical context. It refutes a certain way of thinking about sets that was plausible at the time (sets as extensions of definable properties). If you give an informal description that doesn't stress that context, then it's really hard to see how we're talking about anything other than the liar paradox in other words.
But keeping those concerns in mind, we should be able to come up with something. I hope I've explained what I didn't like about your previous effort, but I'm not suggesting we quit working on it. I don't have a suggested wording at the moment. --Trovatore 18:40, 17 April 2007 (UTC)

What about adding a sentence to the end of the first para: The assumption that sets can be freely defined by any criteria is contradicted by the impossibility of a set containing all sets that are not members of themselves. Such a set qualifies as a member of itself, but at the same time contradicts it's own definition. Diggers2004 10:17, 18 April 2007 (UTC)

That's not too bad. The second sentence needs a little tweaking, I think. And the first might say something like "containing exactly the sets..." rather than "containing all sets...". --Trovatore 19:37, 18 April 2007 (UTC)

Next draft for feedback: "The assumption that sets can be freely defined by any criteria is contradicted by the impossibility of a set containing exactly the sets that are not members of themselves. Such a set qualifies as a member of itself, which then contradicts it's own definition (as a set containing sets that are NOT members of themselves." Diggers2004 00:08, 22 April 2007 (UTC)

I would still like to have a better discussion in the 'informal presentation' section. As a certified non-mathematician, I can't fully follow the (no doubt excellent) formal presentations that follow. I need (and many readers need) something on how, e.g., the barber example resolves if you xxx?? Something to do with a definition (or set) that is really something else, I vaguely remember from a college philosophy class. I'm sure you real mathematicians here could supply what I (and others) seek on this. Thanks. Pechmerle (talk) 21:31, 2 October 2008 (UTC)

## Fuzzy logic

Shouldn't the "Independence from excluded middle" section include something about how the paradox is resolved if fuzzy sets are used instead? (See my previous comment under "Fuzzy Russell".) --SurrealWarrior 02:40, 12 April 2007 (UTC)

Frankly, no. Certainly not without a reliable source. Even then it would be more about the liar paradox in general than about Russell, which, as I mentioned, is specifically a refutation of Fregean set theory. I've never heard of anyone making Fregean set theory work, as a foundation for mathematics, by using fuzzy logic. --Trovatore 03:08, 12 April 2007 (UTC)
But it does work. voidnature 04:58, 15 May 2011 (UTC)
See "Fuzzy Russell" for my reference of a reliable source, the reference has a discussion on a very similar point. K2kingy (talk) 12:54, 17 April 2012 (UTC)

## Confusing equations in Formal derivation

The equations in the section Formal derivation are quite confusing. I tried to convert them to TeX markup, but could not entirely understand the equations in their present form (HTML entities). What is the meaning of the &harr; or \leftrightarrow symbol (↔ / $\leftrightarrow\,\!$)? Why is there so much apparent duplication in the equation?

Web-Crawling Stickler 21:26, 28 April 2007 (UTC)

The remark refers to the following bit, copied from the article:
Definition. $A = \{x : x \in A \leftrightarrow \Phi(x)\}\,\!$ is the individual $A\,\!$ satisfying $\forall x, x \in A \leftrightarrow \Phi(x)\,\!$. All sets are collections, but not conversely.
Presumably the intended meaning of the double-headed arrow here is the logical biconditional operator (denoting "if and only if"). But I agree: also with that explanation I find the notation quite incomprehensible. I would instead expect something like this:
Definition. $\{x : \Phi(x)\}\,\!$ is the individual $A\,\!$ satisfying $\forall x, x \in A \leftrightarrow \Phi(x)\,\!$.
I'm not sure what the function is in this context of the last sentence about collections. Can someone with access to the source this was taken from (Potter, Michael, 2004. Set Theory and its Philosophy. Oxford Univ. Press, pp. 24-25) check if there was some notational mix-up, or else supply an explanation making this notation understandable?  --LambiamTalk 07:23, 5 June 2007 (UTC)
I must say that the section was much more comprehensible before a recent edit that replaced HTML mark-up by LaTeX-style formulas, but also introduced several other less felicitous changes. I think I've restored and improved the section to something presentable.  --LambiamTalk 08:04, 5 June 2007 (UTC)

Haven't read your article, looking forward to it, but let me just say that the Russell paradox is specifically about sets. Not anything else. That's what makes it different from the liar paradox and other self-reference/diagonalization notions. Please don't add any text that muddies that take-away message, because it's a constant battle to keep the distinctions clear. --Trovatore 17:48, 15 November 2007 (UTC)
I have put some comments on the talk page of the new article.  --Lambiam 19:43, 15 November 2007 (UTC)

## Is this a valid example of the paradox and if so, is it worth keeping as such?

Wikipedia:List_of_all_lists_that_do_not_contain_themselves it's up for WP:MFD but I most comments don't seem to understand the concept. It appears to be an example of Russell's Paradox. If so, is it worth keeping? If the answer is yes, you may want to comment at WP:MFD#Wikipedia:List_of_all_lists_that_do_not_contain_themselves (if the answer is no you can too, but that's probably the consensus by now anyway). --Doug.(talk contribs) 03:17, 16 November 2007 (UTC)

• On closer review I see that this is actually the example used in one part of this article. Please comment on the MFD.--Doug.(talk contribs) 06:07, 17 November 2007 (UTC)

Russell’s paradox considers “set of all sets that are not members of themselves”. (BTW “all” is replaced by “exactly” on Wiki page – while it is supposed to have the same meaning it may be misleading to some readers). Such a set appears to be a member of itself if and only if it is not a member of itself, hence the paradox.

Lets start exploring Russell’s paradox by trying to imagine how a “set that IS a member of itself” would look like in reality, shall we?
We know that a “set” is a collection of objects. Let’s assume that we have a collection of beers, not a huge one, just a few bottles. Let’s put them in a refrigerator (aka fridge) to chill. Also, we have a collection of snacks – beacon, salted pork, fish and such – lets put it all in the fridge as well.
Now, as you may realize our fridge is a collection (set) of collections. What Russell’s logic (which is part of naïve set theory of cause) wants us to do is to include this set as a member (and hence, a subset) of itself. In real life this translates into a request to put the fridge with everything it holds into itself. I hope we’ll come to agree that this request just doesn’t make any sense. Neither should such operation be allowed in a consistent theory.
But let’s keep following Russell’s paradox’s logic. We are supposed to collect all the sets that are NOT members of themselves. Trust me, there’s a bunch of those. Collections of stamps, tea mugs, paintings, luxurious cars, bridges, cities, stars in the universe. Man, we’ll need a bigger fridge. But we are not to be afraid of infinites. Let’s just imagine that we’ve bested this enormous task. Now what Russell’s paradox wants us to do is to realize that our fridge that holds all those stuff is a set that is not a member of itself and therefore should be included into itself thus creating the paradox. Yet again, we are asked to put the darn fridge with everything in holds into itself and we already know that this just ain’t gonna happen.
Thus in my understanding, Russell’s paradox arise due to performing an illegal (non-sense) operation that should not be allowed in a theory to be consistent.
--Orlangoor (talk) 15:13, 16 April 2008 (UTC)

What you seem to be arguing is that the Russell paradox should not be surprising. That's fine. Not too many mathematicians or logicians really think it is surprising, these days, with benefit of the developments in the intervening hundred years. But matters were different at the time. Gottlob Frege had proposed a project to reduce all of mathematics to logic alone (the so-called logicist idea), starting with his interpretation (probably misinterpretation) of Cantor's set theory. The Russell paradox completely demolished Frege's approach. (It didn't destroy logicism completely, because it didn't rule out that mathematics might be reduced to logic alone in some other way. Many, but by no means all, mathematical logicians consider that possibility to have been ruled out by Goedel's incompleteness theorems.) --Trovatore (talk) 21:38, 25 April 2008 (UTC)
Well, I do believe sets can contain themselves. Read the natural numbers article. Now take the construction of natural numbers in set theory where S(a)=a ∪ {a}. Then, lets start with ∅. Then, keep making sure "this set contains itself". You end up with $\aleph_0$. Is $\aleph_0$ consistent? What do you think? voidnature 12:07, 14 May 2011 (UTC)

## Huh?

I know what I'm about to write may not endear me intellectually to my more intelligent superiors but this is one of a stream of articles I have been trying to read of the last few days which I can only gain a mild understanding of the subject from. I know that this is an encylopedia, but IMO (perhaps that dosen't count at all) encyclopedias are not intended to give an in depth mathy or sciency discussion or description of the subjects that they deal with, but an accessible overview. Perhaps I am just very stupid, but this article seems quite hard to understand, and the reason for this seems to be that whoever is writing this is being either unecessarily prolix in their description in order to show off they big brain, or that the people who are writing this are so far above the average mortal they don't seem to understand that people may be casually intrested in this subject and may not have a Phd in mathematics. The technical jargon here is all linked, but whenever I find I embarck on a journey to find the meaning of a mathematical term I have never seen before, I find five others on its article page, all described in such an unecessarily long-winded manner as Russel's paradox. If some sympatheitc demigod of the mathematical world is out there, would you take heart and see that stupid people such as myself would like to understand these problems, but would also like to do this without having to take a university course. Perhaps a way around this, as I have seen on some of the chemistry-related articles (a subject with which I am better acquainted with) would be to give the lead and most of the article in simple English (i.e. using terms that may see light outside the maths deparment at a Uni), and create sections where an in-depth knowledge can be granted. I once again humbly apologize for my stupidity.86.138.248.126 (talk) 18:38, 4 July 2008 (UTC)

Chalk it up to the wiki process. It is easier to add a fact to an article than it is to organise a collection of facts into something readable. If you can give an example of an article on a chemistry topic (or any topic) which is of comparable difficulty but which is much more accessible, it would be appreciated. --EmbraceParadox (talk) 19:47, 4 July 2008 (UTC)

Third law of thermodynamics might be comparable, but I think Russel's paradox is on a different level, still I find the article on the law easier to understand. I thik the problem is that this article is very well written, but not in a way that is intended to explain the maths to people who don't know much about it. At the same time, I understand if you don't know anything about maths going straight here would be quite foolish, still I think a better balance could be struck, because I guess the point of an encylopedia is to explain things, not just to state them as they are.86.138.248.126 (talk) 23:54, 4 July 2008 (UTC)

I think Russell's paradox should actually be easier to understand than the third law of thermodynamics, so the example is good. Not much should be required to at least get what it is and why people cared at the time, even if ZFC is too abstruse. --EmbraceParadox (talk) 18:20, 5 July 2008 (UTC)

Looking through this page for the first time I was surprised to discover that the popular version of the paradox in terms of the famous barber is not given more prominence in the lead. It is true that, as noted later on, it may lead to a false impression that the reader can easily refute the paradox. However, this concern can be immediately addressed by pointing out that the true significance of the paradox lies in a re-examination of the foundations of "naive" set theory usually ascribed to Frege, which was found by Russell to be lacking in coherence. I may get around to barberizing the lead unless there is staunch opposition to such a move. Katzmik (talk) 09:00, 5 August 2008 (UTC)

Look, there are all sorts of closely related paradoxes, but the distinctive feature of Russell's paradox per se is that it's specifically about sets. The Spanish Barber is not about sets, at least not directly. Yes, you can rephrase it in terms of sets (or sets-as-properties), and then it becomes Russell's paradox, but that's not its origin. I really think the focus in this article should be on sets, and therefore not on the related paradoxes that aren't about sets. --Trovatore (talk) 18:56, 5 August 2008 (UTC)

OK, well, that's staunch opposition if there ever was such :-) If there is an appropriate forum where this could be discussed, perhaps it could make for an interesting discussion. wikiproject:math seems a bit too distant. Was the barber paradox used by Russell himself or not, and if not who used it first as a popular explanation for Russell's paradox? A lead to a wiki article in general tries to make things as accessible as possible, even if only as a first approximation, and I find the current lead forbidding. Katzmik (talk) 10:45, 6 August 2008 (UTC)

You can modify its form; some forms of modification are valid and some are not. I once had a form suggested to me which was not valid, namely the question whether the barber shaves himself or not. You can define the barber as "one who shaves all those, and those only, who do not shave themselves." The question is, does the barber shave himself? In this form the contradiction is not very difficult to solve. (Bertrand Russell, "The Philosophy of Logical Atomism", reprinted in The Collected Papers of Bertrand Russell, 1914-19, Vol. 8., p. 228.)
(my emphasis by underlining)  --Lambiam 19:42, 12 August 2008 (UTC)
Thanks for bringing this to my attention. It may be worth it to include this comment at Barber paradox. Katzmik (talk) 10:25, 13 August 2008 (UTC)

## A little note on Russell's paradox: a set can't contain itself

--Faustnh (talk) 21:20, 12 December 2008 (UTC)

The construction "this set contains itself" should be considered harmful (erroneous, incorrect...).

Let's see. Imagine we have the set S. Let's represent it as:

S

Imagine we wonder now what is S (what components it owns), and imagine we are answered S contains itself. Imagine we represent this as:

S
S

But now we could wonder about the second appearance of S (the included S). So we would remember again the definition we were given ("S contains itself"), and we would write:

S
S
S

Now we could wonder about the third appearance of S (the new appearance recently added), we would remember the initial definition, and we would do:

S
S
S
S

Obviously we are in an endless undetermined process (just think about languages and metalanguages, or about dictionary circularities).

(Don't confuse this with recursion; recursion is not undetermined).

Look, it's certainly true that a set as studied in contemporary set theory — that is, an object that shows up at some stage of the von Neumann hierarchy — cannot be an element of itself, and this makes perfect sense at an intuitive level, if your intuitive sets are the objects from the von Neumann hierarchy.
But — and this is the central point — those are not the objects Frege thought he was describing. Frege thought of sets as being extensions of (presumably definable) properties, and that they were more or less the same things as the formulas defining them. That was the central idea of his logicist approach. And for this Fregean concept of set, there's no clear reason a set shouldn't have itself as an element, if it satisfies its own defining property.
What the Russell paradox shows is that that approach doesn't work. This is the context in which it needs to be understood. Discussions of the RP divorced from the historical Fregean context always have this pointless quality to them. --Trovatore (talk) 01:53, 13 December 2008 (UTC)

--Faustnh (talk) 14:13, 13 December 2008 (UTC) . I understand that Russell's paradox and Russell's suggestions just appointed that sets can't contain themselves. Russell's work just wanted to assume sets can't contain themselves.

No, I am afraid you have not understood correctly. There's a lot more to it than that. --Trovatore (talk) 19:39, 13 December 2008 (UTC)

--Faustnh (talk) 19:51, 13 December 2008 (UTC) . Principia Mathematica. Best regards.

Hey, read the natural numbers article. Now take the construction of natural numbers in set theory where S(a)=a ∪ {a}. Then, lets start with ∅. Then, keep making sure "this set contains itself". You end up with $\aleph_0$. Is $\aleph_0$ consistent? What do you think? voidnature 10:09, 14 May 2011 (UTC)

## R or its mirror image?

I've always seen {{unsourced}} this R set defined with an inverted R (the mirror image of it). Maybe the article should use the mirror image too. Albmont (talk) 14:33, 14 May 2009 (UTC)

## Frege vs Cantor

I just quickly wanted to change "Frege" into "Cantor" in the lead paragraph - wondering why this oddity was never spotted by anyone I went through page history and saw that similar changes were indeed reverted in the past.

Now, from as far as I remember, Russell wrote this famous response to Frege in his review of the Begriffsschrift. But the theory that was afterwards dropped was not formal propositional logic as Frege drafted it, but Cantor's set theory.

As I hate being reverted I would like to discuss this here ;) --Pgallert (talk) 17:54, 10 August 2009 (UTC)

I have massively emended the lead para and history to reflect the available documentation in particular van Heijenoort (1967) From Frege to Goedel: A Source book in Mathematical Logic, 1879-1931, Harvard University Press, Cambridge MA) and Bertrand Russell's (1903) Principles of Mathematics which is available for download at Googlebooks. I have a tertiary source (Mario Livio (2009) Is God a Mathematician? Simon and Schuster, New York NY) that discusses the paradox pp. 187-191, but Livio states at page 186 that in the context of Frege's Basic Laws of arithmetic -- and this context is important -- "In particular, one of his axioms -- known as Basic Law V--proved to lead to a contradiction and was therefore fatally flawed. ¶ The law itself stated innocently enough that the extension of a concept F is identical to the extension of concept G if and only F and G have the same objects under them. But the bomb was dropped on June 16, 1902, when Bertrand Russell (figure 49) wrote a letter to Frege, pointing out to him a certain paradox that showed Basic Law V to be inconsistent [etc].(Livio 2009:184-187). Thus, in the context, Livio is saying that Russell was responding to Basic Law V, but in fact Russell's own letter that appears in van Heijenoort 1967:124-125 specifically states that the objection lies in Frege's Begriffsschrift. I cannot imagine how Livio, who lists van Heijenoort 1967 in his bibliography, came to this conclusion (or the contextual jump missed him). This is a pristine example of why investigation of primary sources is often mandatory. Bill Wvbailey (talk) 18:11, 9 September 2009 (UTC)
I just found something significant in Bertrand Russell (1920) Introduction to Mathematical Philosphy, Dover reprint 1993 ISBN 0-486-27724-0 (pbk). He is discussing fallacies that he calls "confusion of types". He says he doesn't want to discuss types here because its too big a topic, but he says that "This necessity [for types] results, for example, from the " 'contradiciton of the greatest cardinal' [here he discribes it]. When I first came upon this contradiciton, in the year 1901, I attempted to discover some flaw in Cantor's proof that there is no greatest cardinal, which we gave in chapter VIII. Applying this proof to the supposed class of all imaginable objects, I was led to a new and simpler contradiction, namely, the following: -- [wherein he discusses "classes that are not members of themselves]" (pages 135-136).
Nowhere does he mention Frege in this context. In his letter to Frege 16 June 1902 he states at the outset that he had had Grundgesetze der Arithmetik for a year and a half but "it is only now that I have been able to find the time for the thorough study I intended to make of your work" (van H 1967:124). He goes on a bit and then brings up "(§9 of your Begriffsschrift)" and notes his "difficulty" with it. (van H 1967:124).
While his 1903 chapter X THE CONTRADICTION is discussed in class-theoretic terms, he devotes an entire appendix A to The Arithmetical and Logical Doctrines of Frege and immediately after an entire appendix B to "The Doctrine of Types". He ends appendix A with the hastly-added note (after corresponding with Frege in June 1902):
Note. The second volume of Gg., which appeared too late to be noticed in the Appendix, contains an interesting discussion of the contradiction(pp. 253-265), suggesting that the solution is to be found by denying that two propositional functions which determine equal classes must be equivalent. As it seems very likely that this is the true solution, the reader is strongly recommended to examine Frege's argument on the point" (boldface added, Russell 1903:522.
Immediately Russell launches into Appendix B with "497. THE doctrine of types is here put forward tentatively, as affording a possible solution of the contradiction; but it requires, in all probability, to be tranformed into some subtler shape before it can answer all difficulties" (Russell 1903:523).
My conclusion from this is that Russell had by 1908 discovered two paradoxes (the Cantor paradox and the "Frege paradox"); one might argue that the two are the same, or not. Perhaps he discovered the "Frege paradox" first after reading "Gg" and then hunting back into Begriffsschrift for the source of it. Perhaps not. Whatever the case, I can see now the source of the contention re Frege vs Cantor. They both represent half the story. And Russell's own memory/thinking probably was changing as time passed. Bill Wvbailey (talk) 20:45, 9 September 2009 (UTC)
The article referenced in the footnote i.e. "Link (ed.)" is excellent. But ends with Russell's hypothesized dismay at Frege's confusion -- Russell hoping that the matter would go away easily with a solution from Frege:
“It is with the letter to Frege that the paradox begins its published history. So it is at this point that the real history begins.”
I found this article ambiguous as to the impetus behind Russell's "paradox(es)" especially toward the end of the article. Bill Wvbailey (talk) 22:41, 9 September 2009 (UTC)
OK, first, to Will: I'd like to look over your changes when I have more time. One thing I notice right off is a couple of problems with the first sentence. First, it isn't a sentence, because it has no predicate — it's actually just a noun phrase with a relative clause or two attached. Second, I don't like the bit about Part of fundamental mathematics, by which I think you really mean foundations of mathematics.
To Pgallert: Why do you say Cantor? It is not clear that the Russell paradox poses any problem at all for (the later versions of) Cantor's informal set theory. (Admittedly this depends on how one reads Cantor, and there is substantial disagreement among scholars on this point.) --Trovatore (talk) 04:13, 10 September 2009 (UTC)
Yeah the first sentence is a mess. Nothing but a clause, and a jumble at that. I saw it yesterday but then forgot to fix it. (Or couldn't figure out a fix . . . After I saw your post and I looked at it again, it resisted my efforts). The fundamental mathematics was there originally. I'll change to foundations of mathematics. The only parts that I've changed are the history and the lead sentence. I could put some of the footnote information into the text. Thoughts? Bill Wvbailey (talk) 13:45, 10 September 2009 (UTC)
Fixed. Am concerned about the 2nd paragraph of the lead -- too much emphasis on sets?: Russell's letter to Frege frames the paradox(es) in logicistic terms first (a predicate predicating of itself) and then in set theoretic terms (the class of classes belonging to themselves). Bill Wvbailey (talk) 15:02, 10 September 2009 (UTC)
Will, I'm afraid I think things are getting less readable, and too "historical". History is important and interesting but shouldn't dominate, especially in the lead section.
Also I think the former emphasis on sets was entirely appropriate. There's really nothing novel about the Russell Paradox in terms of logic — the liar paradox had been known for millennia. The paradox comes into its own because of Frege's attempt to reduce sets to logic. --Trovatore (talk) 20:11, 13 September 2009 (UTC)
Russell clearly framed his letter to Frege in logicistic terms and then in set-theoretic terms. In his 1903 you see the same dichotomy. From what I can gather from the history of Russell's logicism this is about where the foundational split occurs. It's true that for the most part he framed his 1903 in set-theoretic terms, but the quote in the history section shows that his considerations were also logicistic. The logicistic form of the paradox informed his later work; while in his 1919-1920 Introduction to Mathematical Philosphy" he continues to frame it in set-theoretic terms (as I noted above), but he also intends to eliminate "classes" as a "primitive idea" (p. 181), subsituting "the defintion of descriptions . . . which will assign a meaning that altogether eliminates all mention of classes form a right analysis of such propositions . . . classes are in fact, like descriptions, logical fictions, or (as we say) "incomplete symbols". ... The first thing to realize is why classes cannot be regarded as part of the ultimate furniture of the world " (p. 181-182 -- this is quite a bizarre chapter). In other words, his considerations of the paradox from a logicistic point of view are very important to the history. True, Russell failed and the set-theoretic won out. But I'm only interested in getting the historical section right. Bill Wvbailey (talk) 03:00, 14 September 2009 (UTC)
• Trovatore has reverted back several days worth of edits. These include edits I made which added some sourcing to the article. A blanket reversion of multiple edits and editors does not seem satisfactory. As Trovatore seems mainly concerned with the lead, it would be best to hash that out separately. Colonel Warden (talk) 21:23, 13 September 2009 (UTC)
Fair enough -- putting back the lead only for now. However I don't really agree with your reasoning about the blanket reversion. I have difficulty following these changes closely during the work week, and I think they need more discussion. This is in the spirit of WP:BRD. --Trovatore (talk) 21:34, 13 September 2009 (UTC)
• I have no particular axe to grind and was just doing some cleanup. As a general point, the article prior to recent changes had few inline citations and so fell well short of current Wikipedia standards. The process of adding inline citations may help in bringing out and addressing any substantive issues which there may be. And, by studying the sources, we will get a feel as to the appropriate balance between an explanation of the paradox, the history of its development and the state of current work. Colonel Warden (talk) 22:10, 13 September 2009 (UTC)

At this point, I'm after the facts, not an interprettion of the facts. Russell's written words to Frege are the facts. So I reinserted the footnote into the lead. Bill Wvbailey (talk) 22:41, 13 September 2009 (UTC)

I have no great objection to it as a footnote.
But Will, in general you overemphasize primary sources. Wikipedia, as a tertiary source, is supposed to rely mainly on secondary sources; primary sources are permitted as a backup or for elaboration, but should never be the principal source.
Still, I'm content to have this in the notes section; there it could have some value. --Trovatore (talk) 22:52, 13 September 2009 (UTC)
I'm sure you're aware that the major source of much of this, van Heijenoort, begins all original papers with detailed commentary, sometimes written by him and sometimes by Quine, and others as well. Notably, he had personal access to many of the folks whose papers he printed, including Russell. (This is also true for Davis when I use him). I'm just finding good quotes from the primary sources and then connecting the dots. I'm using other secondary and primary sources as well. But I am doing my best to check the accuracy of the secondary sources. Some of these have turned out to be not so good. And tertiary sources are proving, for the most part, to be just plain abysmal -- incomplete, often shamefully useless (unless they have bibliographies, and many don't). And Russell for his part was an awful scholar -- he rarely footnoted or referenced anything (about the only exception I've found is the introduction to his 1925 Principia Mathematica). Bill Wvbailey (talk) 03:00, 14 September 2009 (UTC)

Here are some of my thoughts about the article as it stands.

Footnote #1 in the present version of the article [1] is way too long for the lede. The point of the lede is to give a general overview of the article topic, not to go into depth about sources. In principle, the things mentioned in the lede are all covered in greater depth lower in the article, and that is where detailed footnotes can be used. The same holds for the long quote from Russell's letter: this is the sort of thing that belongs lower in the article, not in the lede. The lede should really be able to stand alone as a précis of the whole article.

As for the article as a whole, the main issue I see is that essentially all that is discussed is very early history. If we had proper coverage of more recent philosophical literature on the paradox, the history sections would fall in to place more naturally. But without any more recent perspectives, we have an article where the early history dominates.

Finally, it looks to me like the section "Russell-like paradoxes" is a case of misplaced focus. Yes, one can make this simple generalization, but is it of interest to anyone in the literature, or just something an author here thought of one day? — Carl (CBM · talk) 01:02, 14 September 2009 (UTC)

RE a long quote in the lead's footnote: I agree that this is a lousy place for an important quote. But Trovatore doesn't want it overtly in the lead. Okay. So I put it back into the footnote -- I want it somewhere. It shows both the logicistic and the set-theoretic (what? split? trauma, difficulty ... ) repercussions. I could put it into the historical section. The dual consideration is hinted at by the 1903 quote that is there. If I can summon up the strength I'll try to put the quote there. But someone will certainly bitch about the section's "bloat".
The first sentence is flat-out misleading. But I wash my hands of this. I've move the footnote to the history section, tidied up the section. And that's it for me on this, I'm fed up. Wvbailey (talk) 14:19, 14 September 2009 (UTC)
In what way is the first sentence misleading? Above you yourself say that Russell identified the paradox in Frege's Begriffschrift, and also recovered it from an analysis of Cantor's proof that there is no largest cardinal, resulting in a contradiction if there is a "supposed class of all imaginable objects" (the latter class not being part of Cantor's later ontology, surely, given that he had already refuted its existence by proving there is no largest cardinal, but whose existence would follow from Frege). --Trovatore (talk) 18:10, 14 September 2009 (UTC)
Here's why. From what I can tell (esp. that paper in the "100 years of Russell's paradox"), Russell did the "recovery" from Cantor's naive set theory first, and then was stunned to find an antinomy in Frege's logic, and then after a lot of noodling felt he could equate them. This is what the first sentence says. It may not be its intent, but it's what it says:
"In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Frege leads to a contradiction."
Frege didn't construct a set theory, at least per my cc of the Begriffschrift that shows nothing but logic diagrams. Russell 1903 says:
"481. The word Begriff is used by Frege to mean nearly the same thing as propositional function (e.g. FuB. p. 28)-; when there are two variables, the Begriff is a relation." van H says: "he wants to employ a logic in order to provide a foundation for arithmetic".

Livio 2009 says: "Around the same time that Frege was developing his logicist program, the Italian mathematician and logician Giuseppe Peano ws attempting a somewhat differnt appproach ... The next step was taken by Bertrand Russell. Russell maintained that Frege's original idea-- that of deriving arithmetic from logic -- was still the right way to go" (p. 188). And van H says, about Zermelo's 1908, "This paper presents the first axiomatic set theory. Cantor's definition of set had hardly more to do with the development of set theory than Euclid's definition of point with that of geometry" and then van H invokes Dedekind. I don't see any mention of "Frege's set theory" in Zermelo 1908; in fact, no mention of Frege at all. I haven't seen anywhere any references to "the naive set theory of Frege". By identifying the two -- Cantor's ordinal business with a paradox from Frege's logical ideography, you've made an intellectual jump not supported by the meager sentence at hand. In fact I think the identification process is probably pretty sophisticated -- identifying "class" with the ideographic symbol for "relation" or something to that effect. Readers will be misled: "Oh, I didn't know Frege was responsible for "set theory"." To be fair, the sentence should have a "source?" tag stuck on it. Bill Wvbailey (talk) 19:21, 14 September 2009 (UTC)

Um. I had always understood that Frege believed he was formalizing Cantorian set theory (but got it sort of wrong). I admit I haven't ever actually read Frege. If it's true that Frege was not talking about sets, then I agree that's a problem with the current text. But I'm skeptical.
The problem with localizing the issue in "Cantor's naive set theory" is that it suggests that informal Cantorian set theory was inconsistent, which really isn't true, at least not the later Cantorian set theory. This is a point made most forcefully by Wang Hao; Frapolli later criticizes Wang, but makes a distinction between the two separate Cantorian theories. --Trovatore (talk) 20:00, 14 September 2009 (UTC)

---

RE About the modern history: My guess is the value of Russell's "paradoxe(s)" are (i) cautionary, (ii) historical -- primarily to be connected with foundational issues of the very early 20th C. From the point of view of the "practical arts", in the 1950's we engineers discovered ways of analyzing truth tables so that we could intentionally introduce impredicativity. Then we domesticated the resulting paradoxes, and we put them to work. We turned them into oscillators and flip flops (e.g. set-reset memory, type D flip-flops etc) and when we felt ambitious, state machines. (Quine and McClusky were behind this, but I'm not sure of the historical details). I'm not aware of any further progress on that front (no need, matter resolved). From the theoretical-mathematics side I've seen references in van Heijenoort to resolving Frege's "way out". But these occurred in the 1950's-1960's. Bill Wvbailey (talk) 02:32, 14 September 2009 (UTC)

## all propositions

We shall, therefore, have to say that statements about “all about “all propositions” are meaningless (Whitehead and Russell 1910, 37) This is a proposition about all propositions about all propositions. It declares itself to be meaningless.

No statement about all members of a group can be true.

The above statement is about all members of the group of statements about all members of a group. The statement itself is a statement about all members of a group. It is a member of the group it is referring to. It is a self-reference statement that declares itself to be untrue.Davidsstorm (talk) 16:14, 19 December 2009 (UTC)

Ha! You sure? Let's see, "all members of the group of statements about all members of a group is equal to itself". That was all about "all members of the group of statements about all members of a group". It is obviously true. What do you think? voidnature 11:55, 14 May 2011 (UTC)

## The cause of Russell's paradox?

P = {all sets which contain themselves} Q = {all sets which do not contain themselves}

A set is defined as a collection that contains all the elements of certain properties. Normally, the properties of an element can be written as an expression just as (x + y + ...), when the expression is true the element belongs to a set, and when the expression is false the element not belongs to a set.

In Russell's paradox, P, Q are not existent sets. It is just as a question that in the conditions of Russell's paradox, whether there is a reasonable solution for P, Q. That is, Russell's paradox using a set of constraints to describe the set, a constraint can be written as a logical equation, so that multiple constraints can be written as a logical equations. Due to a logical equations may has no solution, when the constraints contains a description of whether an element belongs to a set, and the equations has no solution, then can not determine whether the element belongs to a set or not.

The direct cause of Russell's paradox is the definition of element properties conflict with the default rule of set theory, "an element has certain properties equal to the element belongs to a set". That is in the definition of element properties, it may different between "have some property" (the value of the expression x + y + ... ) and "belongs to a set" (a variable in the expression x + y + ... ). The value of the expression and a variable in the expression may not the same.

Qiuzhihong (talk) 17:16, 20 October 2010 (UTC)

## Russell's Paradox Only Asserts That {x|x∉x} Does Not Exist

All Russell's Paradox says is that defining {x|x∉x} is paradoxic. Since defining {x|x∉x} is paradoxic, {x|x∉x} cannot be defined. When people say "any definable collection is a set" and "Russell's Paradox showed that the naive set theory created by Georg Cantor leads to a contradiction" they should notice the paradox itself asserts that defining {x|x∉x} is paradoxic, therefore is not a definable collection, therefore is NOT a set. They should then notice the problem is NOT with the naive set theory, but with idioticy of people saying "any definable collection is a set" but defining {x|x∉x} is paradoxic and this is done in naive set theory so naive set theory is not consistent. So what do you think? voidnature 10:00, 14 May 2011 (UTC)

I hope you are right! Funny most people have been either pointing how this paradox is similar to all others or responding saying its diferent because of it's context. Now my humble understanding of logic tells me paradoxes always look similar (A is equal to itself and to its negation). The relevant diference is the falacious argument(s) that led to the paradoxic conclusion. If you are right (I am no expert in mathematics and less even in set theory), I would sugest the article to be rewritten as not only you focused the point but you also state it in a language a dumb like me can understand.Learningnave (talk) 12:19, 30 June 2011 (UTC)

## Edit of 4 july 2011

Per BRD I'm moving this here. It looks like self-promotion of a paper. If others agree to it, it needs work on its English:

Important note: Restriction on the scope of diagonal argument is set using two absolutely different proof techniques. Along with this restriction one of the proof techniques analyzes contradictory equivalence (R ∈ R ↔ R ∉ R) in a rather unconventional way and resolves it. Cantor’s and Russell’s paradoxes are resolved. https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B_tihhgZ1L4wMjUxZDI5NjUtYTZlMy00NDJhLWJjN2MtMDAzNDUzOWQ2Y2Ew&hl=enThe paper is in English (according to a professional mathematician having position in USA - quite readable English), though, of course, it is not English of an English speaking person.

So many efforts were made by many people to work out ways around the paradoxes. The thing is done now. They are resolved. All needed now is just public support.

Thanks a lot in advance, guys!

BillWvbailey (talk) 14:48, 4 July 2011 (UTC)

With regards to the English: unfortunately I couldn't read the paper because I can't read Russian, so I can't determine the author's intent. But what I've done is guess at what the editor is trying to say, above.
But the first sentence does not connect logically to the second sentence. What is missing is an implication that the passive voice expresses. I see two modus ponens, the first one's conclusion applied to the second one:
(1) (P1 & P2) & ((P1 & P2) --> restricted diagonal argument) | restricted diagonal argument;
(2) analysis & ((restricted diagonal argument & analysis) --> resolution) | resolution
"By use of two absolutely different proof techniques, the scope of the diagonal argument can be restricted. Together with this restriction, analysis with one of the proof techniques resolves the contradictory equivalence (R ∈ R ↔ R ∉ R) in an rather unconventional way."

BillWvbailey (talk) 20:39, 4 July 2011 (UTC)

The article does not give much context to the paradox. As a non mathematician I found it hard to understand the true meaning. The existing informal presentation does not really help me. What I found hard to understand was that we are starting directly from an axiom and deriving a contradictory case. Not just making a self contradictory statement.

I have missed out some "for alls" because for most people they are implicit. So the dumb layperson like myself can easily get confused by them. Also I have avoided using "exists" because lay people dont know what it means. I propose this to be a kind of informal introduction separate from the formal description given.

I hope this doesn't offend any mathematicians out there. Thepigdog (talk) 05:35, 22 May 2013 (UTC)

Hi Thepigdog. We always want to make articles as understandable as they reasonably can be given the inherent difficulty of the topic. But this level of step-by-step exposition is really not on the table. Wikipedia is not a textbook. It's a place to look things up, and if you're missing some background information to understand what you find, sometimes you just have to do some work on your own to go get it.
That's not to say the readability can't be improved. I think it can. Your experience could be valuable here, with the specifics of what you found hard to understand. So maybe that means we need some more historical context, with how set theory was understood at the time. --Trovatore (talk) 18:31, 22 May 2013 (UTC)

OK for a long time I did not get the paradox even though it is quite simple.

Maybe I am just thick, but these were the things that were stopping me from understanding it,

• The informal presentation does not look recognisable to me as Russell's paradox. This meant that it was not usefull to me.
• The descriptions of russell's paradox elsewhere in non wiki documents are also confusing because they make a point of talking about both $P(x) = x \not \in x$ and $P(x) = x \in x$ conditions. This was confusing to me. Clearly the russell's paradox is based on the set condition $P(x) = x \not \in x$.
• Initially I didn't get the difference between a paradox, and a self contradictory statement. I didnt see that we were just applying the axioms to get the paradox, as against adding a new statement that was contradictory.
• The statement of the axiom of unlimited comprehension as :$\exists y \forall x (x \in y \iff P(x))$ is really neat, but I was expecting to see set builder notation. So I didn't recognise it. I know for a mathematician this may seem stupid, but to me, at least initially, something in curly brackets looks like a set.
• The background history of how the mathematicians were axiomatizing maths is very important for understanding the context and importance of the paradox.
• Univeral quantification is really hard for non mathematicians (or computer programmers) to accept. Some explanation of the historical concept, because when you say for all y, I must admit I find it hard to picture what is going on.
• How much memory are you allocating to that variable? Small joke there.
• How can you check each case, at least in principle?
• The article doesnt explicitly say how the axioms were changed in response to Russells paradox. I still dont clearly see the full picture for this.
• Initially I was confused by what does $x \in x$ mean. How can a set be an element of itself.
• History section is really unreadable for the non technical, because it is full of quotes, each one of which is unintelligible until you understand the full context.
• If you have this kind of history section for the mathematician, I think you need at least explain what it all means clearly in lay terms. Otherwise it appears like you are just name dropping about some "in" joke.
• The page uses the term "Set Theoretical" with no lookup to find out what it means. — Preceding unsigned comment added by Thepigdog (talkcontribs) 22:54, 22 May 2013 (UTC)

The other pages related to this like the one on ZFC axioms are also fairly unintelligable. Too many for alls and exists, at least for me.

Anyhow thanks for your patience. If you could just increase the level of explanation in the wiki even just a little bit it would make it much more usable. Sometimes it seems only usefull if you know the stuff elsewhere already. Thepigdog (talk) 22:41, 22 May 2013 (UTC)

### Informal statement

Consider a library which has many catalogs of books. There may also be master catalog, which is a catalog of catalogs. A master catalog might include an entry referring to itself. Now suppose that each catalog has a property, or membership condition, which a book or catalog must satisfy to be listed in the catalog.

Suppose the membership condition for a catalog is that the catalog is not a member of itself. Call this catalog the "non self members" catalog.

Now ask if "non self members" is a member of "non self members".

• If we say it is, then it fails the membership condition.
• If we say it is not, then it passes the membership condition which shows the catalog is a member of itself.

So "non self members" cannot be in or not in "non self members". Either statement leads to a contradiction.

### Background

In the 19th century, mathematicians wanted to organise mathematics by finding the smallest set of statements from which all other statements could be proved. These statements would be called the axioms.

Axioms should be general statements that are obviously true. In choosing axioms mathematicians had to make sure that the axioms were true in all cases. A single false axiom would allow anything to be proved, and make the whole of mathematics meaningless.

They started by building up the things which the axioms describe, the positive integers and the sets. So if I start with the number $0$ in the integers and define a successor function $S(n)$,

$x \in \mathbb{N} \implies s(x) \in \mathbb{N}$

I can apply this rule as many times as I like to build up the set of positive integers. This idea of "as many times as I like" came to be called infinity.

Mathematicians like to make general statements, so they like to make statements that are true "for all" values. So we can say,

$\forall x \in \mathbb{N} (x >= 0)$

The use of "for all" is called quantification. In the above statement x is quantified over the positive integers. This means that we can easily show the statements is true in a finite number of cases, and we can also define a process that proves the statement for as many positive integers as you want.

A paradox is an apparently sound series of logical deductions that leads to an impossibility. For a set of axioms, a paradox is a the application of the axioms so as to arrive at a contradiction.

A paradox should be distinguished from a self contradictory statement, or a falsehood. A statement is self contradictory if, in it's own statement, implies that it is false. For example "This statement is false".

A falsehood is a statement that is not true, or that contradicts the axioms in the axiom set. For example $2 * 2 = 3$.

Although a paradox is superficially simmilar to a self contradictory statement, a paradox is only resolved by rejecting or changing the set of axioms used in constructing the paradox.

### Axiom of Unrestricted Comprehension

When building up the axioms of set theory the following axiom of Unrestricted Comprehension was considered,

$\{x \mid P(x)\}$ is a set

This is a different kind of statement than the one above. There is no domain for $x$. We say that $x$ is universally quantified.

To explain the meaning of this notation, $\{x \mid P(x)\}$ is the name given to a set for which each element satisfies P.

$z \in \{x \mid P(x)\} \iff P(z)$

Lets look at example,

$\{x \mid x \in \mathbb{Z} \and x * x = 4\}$

We know that this is $\{2, -2\}$. But how do we know? We can go and test in a finite number of cases. And then we can construct a process that will prove as many of the integers $> 2$ or $< -2$ do not satisfy $x*x =4$.

But what if we remove $x \in \mathbb{Z}$

$\{x \mid x * x = 4\}$

Now we dont know where to start to prove the statement. This made some people question the axiom.

The axiom or unrestricted comprehension allows us to choose any $P(x)$. Russell chose the predicate,

$P(x) = x \not \in x$

This is a weird but seemingly harmless choice. The axiom says any function of $x$ can be chosen so,

$\{ x \mid x \not \in x \}$

Lets call it $Y$.

$Y = \{ x \mid x \not \in x \}$

so now test if $Y$ is in $Y$. Substitute $Y$ for $z$, and $x \not \in x$ for $P(x)$ in,

$z \in \{x \mid P(x)\} \iff P(z)$

gives

$Y \in Y \iff Y \notin Y$

This is a statement is always false, which is a contradiction.

Because we only applied axioms to arrive at this contradiction then one of the axioms used here in constructing the contradiction is false. Following the discovery of the paradox the axiom,

$\{x \mid P(x)\}$ is a set

was rejected. It was later replaced with the Axiom schema of specification, which can be informally stated as,

$\forall S \{x \mid x \in S \and P(x) \}$ is a set.

in ZFC set theory. It would be nice to say exactly which axioms replace unrestricted comprehension in ZFC.

## Cantor/Frege again

I guess I hadn't been paying attention to this, but the lead now effectively claims that the Russell paradox refutes Cantor's set theory, which is not at all clear (for example, by Wang Hao's reading of Cantor's theory, it does not). Frege's theory, on the other hand, is directly refuted, but in one of the previous go-rounds, someone claimed that Frege's theory was not exactly a set theory.

Things absolutely cannot stay as they are — Frege must be mentioned, and the lead cannot claim baldly that Cantor's theory has a contradiction. But the exact resolution is somewhat open. Thoughts? --Trovatore (talk) 16:04, 4 June 2013 (UTC)

Here's what happened: The change was made by an anonymous editor in two attempts back in 8 January 2011. The version went from:
"In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory of Richard Dedekind and Frege leads to a contradiction."
. . . to (notice the typos "lGeorg Cant" -- this is corrected in a second edit on the same day):
"In the foundations of mathematics, Russell's paradox (also known as Russell's antinomy), discovered by Bertrand Russell in 1901, showed that the naive set theory created by lGeorg Cant leads to a contradiction".
Per our discussions of this before, I'm not confortable with the labelling of Frege's and Dedekind's works as "naive set theory"; I'd call them "precursors to Russellian logicism". But at the least I'd suggest we restore the sentence to include Frege and possibly Dedekind and remove Cantor.
I'm not really suggesting that Cantor shouldn't be mentioned; the paradox does refute that which I gather Russell and Frege took to be Cantorian set theory. But given Wang's objection, I don't think we should imply that R&F were right about that. I really am having trouble coming up with good wording. Anyone have ideas? --Trovatore (talk) 19:00, 7 June 2013 (UTC)
Rats, I just noticed that this is still a hanging issue. I really want to fix this, and I am not seeing how. It is absolutely not acceptable to assert baldly that RP refutes Cantor's set theory. What it really refutes is Frege's system, and Russell's interpretation of Cantor's set theory. But I'm just not seeing a good way of wording it that fits in the lead. --Trovatore (talk) 19:55, 5 September 2013 (UTC)

## Nitpicking - I cannot see any paradox here.

By definition, "any definable collection is a set", then, as the article and everyone above verbosely points out, the following turns out to not be definable: "Let R be the set of all sets that are not members of themselves", thus by definition, it's NOT a set, so where is the paradox?

It simply appears to me that simplistic "inverse" operations cannot be performed on sets, or that folks are trying to introduce incompatible things into sets - eg: having a "set" (i.e. some definition pertaining to elements) inside a set of elements, (e.g. mixing definitions and values together) is not unlike having imaginary and real numbers together - yeah, sure, do that, but you can't just ignore the fact that those beasts are different things (or different dimensions, if you think about this stuff spatially, like I do).

We all understand "number lines" - if you take any value on that line, and add something positive - you move right, or subtract, you move left. Taking an "inverse" of a set, is like telling someone - here is my new position on the number-line - tell me what I added or subtracted to get here... when the "new position" is neither left nor right - but *up* (i.e. the spacial definition of an imaginary number - sqrt(-1)). — Preceding unsigned comment added by 120.151.160.158 (talk) 07:58, 6 June 2013 (UTC)

## Examples?

Shouldn't this article start out by giving an example of a set that is a member of itself? rowley (talk) 19:44, 5 September 2013 (UTC)

According to the most usual modern conception of set, the so-called cumulative hierarchy, no set is a member of itself. So in that sense, no, there is no such example so we can't give one.
However, it might be useful to give examples of "sets" that would be members of themselves if sets were identified with extensions of predicates (the incorrect conception that is refuted by Russell's paradox). For example, if there were a set of all abstract objects, it would itself be an abstract object, and therefore an element of itself. I'm not against adding such language, provided wording can be found that's not confusing in light of my first point. --Trovatore (talk) 19:50, 5 September 2013 (UTC)