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{{short description|Paradox in the foundations of mathematics}}

factoids
text{Let } R = { x mid x not in x } text{, then } R in R iff R not in R
In 1908, two ways of avoiding the paradox were proposed, Russell's type theory and the Zermelo set theory, the first constructed axiomatic set theory. Zermelo's axioms went well beyond Gottlob Frege's axioms of extensionality and unlimited set abstraction, and evolved into the now-standard Zermeloâ€“Fraenkel set theory (ZFC). The essential difference between Russell's and Zermelo's solution to the paradox is that Zermelo altered the axioms of set theory while preserving the logical language in which they are expressed (the language of ZFC, with the help of Thoralf Skolem, turned out to be first-order logic) while Russell altered the logical language itself.BOOK, A.A. Fraenkel, Y. Bar-Hillel, A. Levy, Foundations of Set Theory, 1973, Elsevier, 978-0-08-088705-0, 156â€“157,

## Informal presentation

Let us call a set "abnormal" if it is a member of itself, and "normal" otherwise. For example, take the set of all squares in the plane. That set is not itself a square in the plane, and therefore is not a member of the set of all squares in the plane. So it is "normal". On the other hand, if we take the complementary set that contains all non-(squares in the plane),{{efn|As opposed to (non-squares) in the plane, which is the complement with the universe of "things in the plane".}} that set is itself not a square in the plane and so should be one of its own members as it is a non-(square in the plane). It is "abnormal".Now we consider the set of all normal sets, R. Determining whether R is normal or abnormal is impossible: if R were a normal set, it would be contained in the set of normal sets (itself), and therefore be abnormal; and if R were abnormal, it would not be contained in the set of all normal sets (itself), and therefore be normal. This leads to the conclusion that R is neither normal nor abnormal: Russell's paradox.

## Formal presentation

Define Naive Set Theory (NST) as the theory of predicate logic with a binary predicate in and the following axiom schema of unrestricted comprehension:
exists y forall x (x in y iff varphi(x))
for any formula varphi with only the variable x free.Substitute x notin x for varphi(x). Then by existential instantiation (reusing the symbol y) and universal instantiation we have
y in y iff y notin y
a contradiction. Therefore, NST is inconsistent.ENCYCLOPEDIA, Russell's Paradox, The Stanford Encyclopedia of Philosophy, 2014, Zalta, Edward N.,weblink Irvine, Andrew David, Deutsch, Harry,

## Applied versions

There are some versions of this paradox that are closer to real-life situations and may be easier to understand for non-logicians. For example, the barber paradox supposes a barber who shaves all men who do not shave themselves and only men who do not shave themselves. When one thinks about whether the barber should shave himself or not, the paradox begins to emerge.As another example, consider five lists of encyclopedia entries within the same encyclopedia:{| class="wikitable" valign="top"
List of articles about people: List of articles starting with the letter L: ...
• List of articles starting with the letter K
• List of articles starting with the letter L (itself; OK)
• List of articles starting with the letter M
...
List of articles about places: List of articles about Japan: List of all lists that do not contain themselves:
• List of articles about Japan
• List of articles about people
• List of articles about places
...
• List of articles starting with the letter K
• List of articles starting with the letter M
...
• List of all lists that do not contain themselves?
If the "List of all lists that do not contain themselves" contains itself, then it does not belong to itself and should be removed. However, if it does not list itself, then it should be added to itself.While appealing, these layman's versions of the paradox share a drawback: an easy refutation of the barber paradox seems to be that such a barber does not exist, or that the barber is a woman and therefore doesn't shave. 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. Note the difference between the statements "such a set does not exist" and "it is an empty set". It is like the difference between saying "There is no bucket" and saying "The bucket is empty".A notable exception to the above may be the Grellingâ€“Nelson paradox, in which words and meaning are the elements of the scenario rather than people and hair-cutting. Though it is easy to refute the barber's paradox by saying that such a barber does not (and cannot) exist, it is impossible to say something similar about a meaningfully defined word.One way that the paradox has been dramatised is as follows:
Suppose that every public library has to compile a catalogue of all its books. Since the catalogue is itself one of the library's books, some librarians include it in the catalogue for completeness; while others leave it out as it being one of the library's books is self-evident.
Now imagine that all these catalogues are sent to the national library. Some of them include themselves in their listings, others do not. The national librarian compiles two master cataloguesâ€”one of all the catalogues that list themselves, and one of all those that don't.
The question is: should these master catalogues list themselves? The 'Catalogue of all catalogues that list themselves' is no problem. If the librarian doesn't include it in its own listing, it remains a true catalogue of those catalogues that do include themselves. If he does include it, it remains a true catalogue of those that list themselves.
However, just as the librarian cannot go wrong with the first master catalogue, he is doomed to fail with the second. When it comes to the 'Catalogue of all catalogues that don't list themselves', the librarian cannot include it in its own listing, because then it would include itself, and so belongs to the other catalogue, that of catalogues that do include themselves. However, if the librarian leaves it out, the catalogue is incomplete. Either way, it can never be a true master catalogue of catalogues that do not list themselves.

## Applications and related topics

As illustrated above for the barber paradox, Russell's paradox is not hard to extend. Take:
Form the sentence:
The er that s all (and only those) who don't themselves,
Sometimes the "all" is replaced by "all ers".An example would be "paint":
The painter that paints all (and only those) that don't paint themselves.
or "elect"
The elector (representative), that elects all that don't elect themselves.
Paradoxes that fall in this scheme include:
• The barber with "shave".
• The original Russell's paradox with "contain": The container (Set) that contains all (containers) that don't contain themselves.
• The Grellingâ€“Nelson paradox with "describer": The describer (word) that describes all words, that don't describe themselves.
• Richard's paradox with "denote": The denoter (number) that denotes all denoters (numbers) that don't denote themselves. (In this paradox, all descriptions of numbers get an assigned number. The term "that denotes all denoters (numbers) that don't denote themselves" is here called Richardian.)

{{notelist}}

## References

{{reflist|30em}}
• {{citation|last=Potter|first=Michael|date=15 January 2004|title=Set Theory and its Philosophy|publisher=Clarendon Press (Oxford University Press)|isbn=978-0-19-926973-0}}
• {{citation|last=van Heijenoort|first=Jean|authorlink=Jean van Heijenoort|date=1967|title=From Frege to GÃ¶del: A Source Book in Mathematical Logic, 1879-1931, (third printing 1976)|publisher=Harvard University Press|publication-place=Cambridge, Massachusetts|isbn=0-674-32449-8}}
• {hide}citation|last=Livio|first=Mario|authorlink=Mario Livio|date=6 January 2009|title=Is God a Mathematician?|publisher=Simon & Schuster|publication-place=New York|isbn=978-0-7432-9405-8
{edih}

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