Exponential distribution

Template:Infobox Bertrand Russell 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 Georg Cantor leads to a contradiction. The same paradox had been discovered a year before by Ernst Zermelo but he did not publish the idea, which remained known only to Hilbert, Husserl and other members of the University of Göttingen.

According to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that it must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. This contradiction is Russell's paradox. Symbolically:

$\text{Let }R=\{x\mid x\in ̸x\}\text{, then }R\in R⟺R\in ̸R$

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 Frege's axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical Zermelo–Fraenkel set theory (ZF).

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

Now we consider the set of all normal sets, R. 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:

$\mathrm{\exists }y\mathrm{\forall }x(x\in y⟺P(x))$

for any formula P with only the variable x free. Substitute $x\notin x$ for $P(x)$. Then by existential instantiation (reusing the symbol y) and universal instantiation we have

$y\in y⟺y\notin y$

a contradiction. Therefore NST is inconsistent.

Set-theoretic responses

In 1908, Ernst Zermelo proposed an axiomatization of set theory that avoided the paradoxes of naive set theory by replacing arbitrary set comprehension with weaker existence axioms, such as his axiom of separation (Aussonderung). Modifications to this axiomatic theory proposed in the 1920s by Abraham Fraenkel, Thoralf Skolem, and by Zermelo himself resulted in the axiomatic set theory called ZFC. This theory became widely accepted once Zermelo's axiom of choice ceased to be controversial, and ZFC has remained the canonical axiomatic set theory down to the present day.

ZFC does not assume that, for every property, there is a set of all things satisfying that property. Rather, it asserts that given any set X, any subset of X definable using first-order logic exists. The object R discussed above cannot be constructed in this fashion, and is therefore not a ZFC set. In some extensions of ZFC, objects like R are called proper classes. ZFC is silent about types, although some argue that Zermelo's axioms tacitly presuppose a background type theory.

In ZFC, given a set A, it is possible to define a set B that consists of exactly the sets in A that are not members of themselves. B cannot be in A by the same reasoning in Russell's Paradox. This variation of Russell's paradox shows that no set contains everything.

Through the work of Zermelo and others, especially John von Neumann, the structure of what some see as the "natural" objects described by ZFC eventually became clear; they are the elements of the von Neumann universe, V, built up from the empty set by transfinitely iterating the power set operation. It is thus now possible again to reason about sets in a non-axiomatic fashion without running afoul of Russell's paradox, namely by reasoning about the elements of V. Whether it is appropriate to think of sets in this way is a point of contention among the rival points of view on the philosophy of mathematics.

Other resolutions to Russell's paradox, more in the spirit of type theory, include the axiomatic set theories New Foundations and Scott-Potter set theory.

History

Russell discovered the paradox in May or June 1901. By his own account in his 1919 Introduction to Mathematical Philosophy, he "attempted to discover some flaw in Cantor's proof that there is no greatest cardinal". In a 1902 letter, he announced the discovery to Gottlob Frege of the paradox in Frege's 1879 Begriffsschrift and framed the problem in terms of both logic and set theory, and in particular in terms of Frege's definition of function; in the following, p. 17 refers to a page in the original Begriffsschrift, and page 23 refers to the same page in van Heijenoort 1967:

31 year-old Systems Analyst Bud from Deep River, spends time with pursuits for instance r/c cars, property developers new condo in singapore singapore and books. Last month just traveled to Orkhon Valley Cultural Landscape.

Russell would go on to cover it at length in his 1903 The Principles of Mathematics, where he repeated his first encounter with the paradox:

31 year-old Systems Analyst Bud from Deep River, spends time with pursuits for instance r/c cars, property developers new condo in singapore singapore and books. Last month just traveled to Orkhon Valley Cultural Landscape.

Russell wrote to Frege about the paradox just as Frege was preparing the second volume of his Grundgesetze der Arithmetik. Frege responded to Russell very quickly; his letter dated 22 June 1902 appeared, with van Heijenoort's commentary in Heijenoort 1967:126–127. Frege then wrote an appendix admitting to the paradox, and proposed a solution that Russell would endorse in his Principles of Mathematics, but was later considered by some to be unsatisfactory. For his part, Russell had his work at the printers and he added an appendix on the doctrine of types.

Ernst Zermelo in his (1908) A new proof of the possibility of a well-ordering (published at the same time he published "the first axiomatic set theory") laid claim to prior discovery of the antinomy in Cantor's naive set theory. He states: "And yet, even the elementary form that Russell⁹ gave to the set-theoretic antinomies could have persuaded them [J. König, Jourdain, F. Bernstein] that the solution of these difficulties is not to be sought in the surrender of well-ordering but only in a suitable restriction of the notion of set". Footnote 9 is where he stakes his claim:

31 year-old Systems Analyst Bud from Deep River, spends time with pursuits for instance r/c cars, property developers new condo in singapore singapore and books. Last month just traveled to Orkhon Valley Cultural Landscape. A written account of Zermelo's actual argument was discovered in the Nachlass of Edmund Husserl.

It is also known that unpublished discussions of set theoretical paradoxes took place in the mathematical community at the turn of the century. van Heijenoort in his commentary before Russell's 1902 Letter to Frege states that Zermelo "had discovered the paradox independently of Russell and communicated it to Hilbert, among others, prior to its publication by Russell".

In 1923, Ludwig Wittgenstein proposed to "dispose" of Russell's paradox as follows:

The reason why a function cannot be its own argument is that the sign for a function already contains the prototype of its argument, and it cannot contain itself. For let us suppose that the function F(fx) could be its own argument: in that case there would be a proposition 'F(F(fx))', in which the outer function F and the inner function F must have different meanings, since the inner one has the form O(f(x)) and the outer one has the form Y(O(fx)). Only the letter 'F' is common to the two functions, but the letter by itself signifies nothing. This immediately becomes clear if instead of 'F(Fu)' we write '(do) : F(Ou) . Ou = Fu'. That disposes of Russell's paradox. (Tractatus Logico-Philosophicus, 3.333)

Russell and Alfred North Whitehead wrote their three-volume Principia Mathematica hoping to achieve what Frege had been unable to do. They sought to banish the paradoxes of naive set theory by employing a theory of types they devised for this purpose. While they succeeded in grounding arithmetic in a fashion, it is not at all evident that they did so by purely logical means. While Principia Mathematica avoided the known paradoxes and allows the derivation of a great deal of mathematics, its system gave rise to new problems.

In any event, Kurt Gödel in 1930–31 proved that while the logic of much of Principia Mathematica, now known as first-order logic, is complete, Peano arithmetic is necessarily incomplete if it is consistent. This is very widely – though not universally – regarded as having shown the logicist program of Frege to be impossible to complete.

In 2001 A Centenary International Conference celebrating the first hundred years of Russell's paradox was held in Munich and its proceedings have been published.

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:

List of articles about people: Ptolemy VII of Egypt Hermann Hesse Don Nix Don Knotts Nikola Tesla Sherlock Holmes Emperor Kōnin

List of articles starting with the letter L: L L!VE TV L&H Leivonmäki ... 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: Leivonmäki Katase River Enoshima

List of articles about Japan: Emperor Showa Katase River Enoshima

List of all lists that do not contain themselves: List of articles about Japan List of articles about places List of articles about people ... 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 at least does not shave (a variant of which is that the barber is a woman). 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 Grelling–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 catalog of all its books. Since the catalog is itself one of the library's books, some librarians include it in the catalog for completeness; while others leave it out as it being one of the library's books is self-evident.

Now imagine that all these catalogs are sent to the national library. Some of them include themselves in their listings, others do not. The national librarian compiles two master catalogs – one of all the catalogs that list themselves, and one of all those that don't.

The question is: should these catalogs list themselves? The 'Catalog of all catalogs that list themselves' is no problem. If the librarian doesn't include it in its own listing, it is still a true catalog of those catalogs that do include themselves. If he does include it, it remains a true catalog of those that list themselves.

However, just as the librarian cannot go wrong with the first master catalog, he is doomed to fail with the second. When it comes to the 'Catalog of all catalogs that don't list themselves', the librarian cannot include it in its own listing, because then it would include itself. But in that case, it should belong to the other catalog, that of catalogs that do include themselves. However, if the librarian leaves it out, the catalog is incomplete. Either way, it can never be a true catalog of catalogs that do not list themselves.

Applications and related topics

Russell-like paradoxes

As illustrated above for the Barber paradox, Russell's paradox is not hard to extend. Take:

• A transitive verb <V>, that can be applied to its substantive form.

Form the sentence:

The <V>er that <V>s all (and only those) who don't <V> themselves,

Sometimes the "all" is replaced by "all <V>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 Grelling–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.)

Related paradoxes

• The liar paradox and Epimenides paradox, whose origins are ancient
• The Kleene–Rosser paradox, showing that the original lambda calculus is inconsistent, by means of a self-negating statement
• Curry's paradox (named after Haskell Curry), which does not require negation
• The smallest uninteresting integer paradox

See also

• Cantor's diagonal argument
• "On Denoting", one of Russell's first attempts at critiquing Frege
• Self-reference
• Universal set

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
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  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
  Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region
  
  Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.
  
  15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.
  
  To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

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• Russell's Paradox at Cut-the-Knot
• Stanford Encyclopedia of Philosophy: "Russell's Paradox" – by A. D. Irvine.

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