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| {{Infobox Bertrand Russell}}
| | == 諺にトラブルの根を行くよう == |
| 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 [[David Hilbert|Hilbert]], [[Edmund Husserl|Husserl]] and other members of the [[University of Göttingen]].
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| According to naive set theory, any definable collection is a [[Set (mathematics)|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:
| | 「DOまたは薫の子供のためにため？ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-15.html カシオ 腕時計 gps] '<br>諺にトラブルの根を行くよう<br>シャオヤンの心の笑顔が、発言はしない悲惨な薫子供やレベルの種類はもちろんのこと、ありません、彼は古代の職業を並べる予感を持って、私は怖いですし、滑らかではありません<br>ゆっくりと街に歩いた小型医療セントで<br>が、シャオヤンのコマンド内の5以来の躊躇後者、バオ泉によると、過去歩いた： [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-14.html カシオ 腕時計 ソーラー 電波] '控えめな場合は、彼は時間を見つけることができるようになります、とシャオヤンあなたが学ぶ？私はあなたが本当にミスを伴うことができるかどうかを確認するために興味があった！ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html カシオ 時計 価格] '<br><br>最後の文、薄い蚊が、それは非常に明確にシャオヤンの耳、中に入るとシャオヤンの言葉を聞いている、また、本当にため息をため息<br>「ああ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html カシオ 腕時計 バンド]。」<br><br><br>シャオヤンはゆっくりと言い訳を探していなかった、彼の頭をうなずいて、彼は明らかに彼女の気質の才能に古代の部族の間で子のステータスを吸っ |
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://a-g.ru/forum/viewtopic.php?f=2&t=157461 http://a-g.ru/forum/viewtopic.php?f=2&t=157461]</li> |
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| | <li>[http://cosplaythai.com/index.php?p=blogs/viewstory/139874 http://cosplaythai.com/index.php?p=blogs/viewstory/139874]</li> |
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| | </ul> |
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| :<math>\text{Let } R = \{ x \mid x \not \in x \} \text{, then } R \in R \iff R \not \in R</math>
| | == は、現在光沢「ファン」醸し出しています == |
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| 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 [[Axiom of extensionality|extensionality]] and unlimited [[set builder notation|set abstraction]], and evolved into the now-canonical [[Zermelo–Fraenkel set theory]] (ZF).<ref name="Tetyana Butler">[http://www.suitcaseofdreams.net/Set_theory_Paradox.htm Set theory paradoxes], by Tetyana Butler, 2006, Suitcase of Dreams</ref>
| | 次に、すぐに、お香のようなダンエッセンスのように、急に充填され、オープン [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html カシオ 掛け時計]<br>「ハハ、私はそれを作った！ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-9.html カシオ 腕時計 チタン] '<br><br>彼の手の中に少しバイアスされた白人男性、彼の背中笑い、ダン「医学」の竜眼の大きさに作られた目の下の点で<br>は、現在光沢「ファン」醸し出しています [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-3.html カシオ 腕時計 電波 ソーラー]。<br>男性のための<br>この失態の笑いに関係なく、誰も被告人の存在に触れていないが、前者の目に見てみると、精錬精錬 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html casio 腕時計 説明書] '医学'教祖ダン7の中間財 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html casio 腕時計 g-shock] '医学'が可能ないくつかのホット、いずれかを約束しているどこに絶対的なVIP待遇である。<br>ダンはすでに後半に入力されます、そのような時期に、錬金術を続け、二人だけ<br>は今、人は本当に別の偉大な能力はもちろん、ある一人一人を持って、この状況は当然、すべての後に、かつての大多数ではない神経の多くの人ができ恥のようなもの、この場合のポイント |
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://bbs.28jk.com/forum.php?mod=viewthread&tid=738792 http://bbs.28jk.com/forum.php?mod=viewthread&tid=738792]</li> |
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| ==Informal presentation== | | == 13399 == |
| Let us call a set "abnormal" if it is a member of itself, and "normal" otherwise. For example, take the set of all [[square]]s in the [[plane (geometry)|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".
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| 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.
| | 罪は微笑んで彼のあご、Lianbuqingyiをうなずいた、フォローアップ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html casio 腕時計 メンズ]。<br><br>葉のフロントヤードは、ここでは現時点では、ここに囲まれたシルエットが多数、雰囲気は特に緊張している [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-7.html casio 腕時計 説明書]。<br><br>ゆったり前庭には、自然がイェ家族や他の人であるが、もう一方の側が、ほぼ道路はピンクのローブシルエットに身を包んで、明確に異なる2つの陣営に分離された、これらの人々はまっすぐ立って、運動量がかなりあることが今曹操ダン地域の位置と、によって、家族や他の人だけでなく、もちろん優越感のようなものの目に見える横暴、反対の葉は確かに、家を比較することはできないままになります [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-15.html カシオ 腕時計 激安]。<br><br>曹操、最前線で歩行者は、2赤いローブ老人、二人の男と無関心の外観、壮大な勢いで満たされた全身で、彼の手には、「挿入」それは、もちろん、人を見て珍しいことではありませんようにスリーブとの間に、それは、ほとんど見え人々が気に持っているので、アイデアは、彼らが持っていることではなく、高齢者の灰色のローブの前で二人の男 |
| | 相关的主题文章： |
| | <ul> |
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| | <li>[http://117.21.174.74/upload/forum.php?mod=viewthread&tid=3219673&fromuid=380545 http://117.21.174.74/upload/forum.php?mod=viewthread&tid=3219673&fromuid=380545]</li> |
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| ==Formal presentation== | | == シャオヤン輝いていたの目に == |
| Define Naive Set Theory (NST) as the theory of [[predicate logic]] with a binary [[Predicate (mathematical logic)|predicate]] <math>\in</math> and the following axiom schema of [[unrestricted comprehension]]:
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| :<math>\exists y \forall x (x \in y \iff P(x))</math> | | また、あなたがそれを行うことができれば、私は小岩この人生、そしてあなたに、一つのことをお願いし、「今日：経験は、そのように、あまりにも多く、ここでストレートメデューサを見つめ、目滞留せず、ささやい害はありません。 [http://alleganycountyfair.org/sitemap.xml http://alleganycountyfair.org/sitemap.xml] '<br>これは懇願の言葉でそう言って、彼女の前で彼の最初の時間ですので、奇妙な魅力メドゥーサ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html カシオ 腕時計 チタン] '混乱'で埋め<br>聞いたが、この瞬間に、理解するので、やや狭め彼女をシャオヤンの狭い目を強制、後者の骨誇りに漠然と感情の種類を気に 'セックス'と、彼女は非常に明確、しかし、今日です<br>念頭に置いて<br>奇妙ゆっくり抑うつ気分は、女王メデューサの光は遠くない昔からの「医学」に懸濁時計本体を見て、法執行の前にアヒルチラリ、言った： [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html カシオの時計] '私はそのショットがあなたを守りたい先生？ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html カシオ 時計 電波] '<br>「ああ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html 腕時計 メンズ casio]。」<br><br>シャオヤン輝いていたの目に<br>、それは良い瞬間の後、あまりにも遠く、遠くない視線Meimouアヒルの法執行機関からゆっくりと部分的なメデューサです |
| for any formula ''P'' with only the variable ''x'' free.
| | 相关的主题文章： |
| Substitute <math>x \notin x</math> for <math>P(x)</math>. Then by existential instantiation (reusing the symbol ''y'') and [[universal instantiation]] we have
| | <ul> |
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| | <li>[http://www.aiao168.com/forum.php?mod=viewthread&tid=248919&fromuid=128011 http://www.aiao168.com/forum.php?mod=viewthread&tid=248919&fromuid=128011]</li> |
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| | <li>[http://tm1.math.uh.edu/search.cgi http://tm1.math.uh.edu/search.cgi]</li> |
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| | <li>[http://www.thecastle-pub.com/cgi-bin/guestbook/guestbook.cgi http://www.thecastle-pub.com/cgi-bin/guestbook/guestbook.cgi]</li> |
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| | </ul> |
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| :<math>y \in y \iff y \notin y</math>
| | == '彼と誰もメインtightlippedロード、シャオヤンの動きを見た == |
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| a contradiction. Therefore NST is inconsistent.
| | 主の口が悲観的な笑顔を誘発する、ゆっくりと言った [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-1.html カシオ スタンダード 腕時計]。<br>トレーニングフィールドにわたって掃引<br>シャオヤンの目、この流血の傭兵グループのメンバーに加えてその家族、少なくとも四から五〇〇の人は、この変更1を持っている、それは彼が4〜500「火の蓮のボトルを考え出す必要があったと言うことです「うわあ、この古い男の食欲は、本当に素晴らしい<br><br>「どうやって？ [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-5.html カシオ 腕時計 スタンダード] '彼と誰もメインtightlippedロード、シャオヤンの動きを見た。<br><br>「非常に良い」シャオヤンは、微笑んで言葉を吐き出すが、それはそれらの血まみれの傭兵グループが、また顔」カラー「グレー、期待の心ながら、インスタント冷たい彼は主表面 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-2.html カシオ腕時計 g-shock] '色'だったすべての家族を作ることですゆっくり打ち砕か。<br>'ケースであること、私たちは妻はあなたが珍しいことではないことを知っているが、あなたは本当に正しい、私の家族と彼をしたい場合、私は彼が家に、それは、ノーバディではない、私はあなたが多くの利点を持っていない恐れている [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-6.html casio 電波時計]。にもかかわらず、残すためにあなたを要求する「<br>ではない、彼はいくつかのためらい上にある場合、彼は、すべての家族主を冷笑 |
| | 相关的主题文章： |
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| | <li>[http://www.uuapen.com/home.php?mod=space&uid=920047 http://www.uuapen.com/home.php?mod=space&uid=920047]</li> |
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| | </ul> |
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| ==Set-theoretic responses== | | == 死手段」 == |
| In 1908, [[Ernst Zermelo]] proposed an [[axiomatic system|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.
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| 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 [[Von Neumann-Bernays-Godel set theory|extensions of ZFC]], objects like ''R'' are called [[proper class]]es. ZFC is silent about types, although some argue that Zermelo's axioms tacitly presuppose a background type theory.
| | つかむ、シャオヤンの体はピンク色「色」ガラス張りの外観に変更すると同時にほとんどであり、その味は突然多くの手段によって、バックハンド高騰されています [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-4.html カシオ ソーラー電波腕時計] '！死手段」<br><br>'轟音 [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-3.html カシオ 腕時計 電波 ソーラー]！'<br>シャオヤンと<br>でも指、アウトピアス膨大なエネルギーの指を指して、ブラックマウンテンロードがた激しく奇妙な指で押す指が、すぐに側面が触れ、自分たちの生活の崩壊を意味するが、多くの理由に依存していた、巨人5が参照折りたたまエネルギーでは、まだヤンがダウンしました攻撃力の魂です [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-0.html カシオ 腕時計 チタン]。<br><br>はシャオヤンが実際に目が [http://www.ispsc.edu.ph/nav/japandi/casio-rakuten-12.html カシオ 時計 電波] '薬'家族、シャオヤンシャンはまた異なった火の助けを必要とする日に、出現するいくつかの驚き「色」を約束している、彼らの攻撃的なフロント、魂ヤンの顔」色は「突然沈没さ要しました、それだけで撃退することができるであろうが、今、反対側は彼らの力のおかげで行い、それらの予備争うことができなければならない、この進歩はあまりにもいくつかの人々を恐怖されています。 |
| | | 相关的主题文章： |
| 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.
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| 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 [[transfinite recursion|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]].
| | <li>[http://www.umanosato.com/clip/clip.cgi http://www.umanosato.com/clip/clip.cgi]</li> |
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| 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]].
| | <li>[http://www.liberalismen.dk/index.cgi http://www.liberalismen.dk/index.cgi]</li> |
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| ==History==
| | <li>[http://ok78.net/home.php?mod=space&uid=71852 http://ok78.net/home.php?mod=space&uid=71852]</li> |
| Russell discovered the paradox in May or June 1901.<ref>{{citation |url=http://books.google.com/?id=Xg6QpedPpcsC&pg=PA350 |title=One hundred years of Russell's paradox |author=Godehard Link |page=350 |year=2004 |isbn=978-3-11-017438-0}}</ref> 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".<ref>Russell 1920:136</ref> In a 1902 letter,<ref>{{citation |url=http://books.google.com/?id=4ktC0UrG4V8C&pg=PA253 |page=253 |year=1997 |title=The Frege reader |isbn=978-0-631-19445-3 |author=Gottlob Frege, Michael Beaney}}. Also van Heijenoort 1967:124–125</ref> 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 (mathematics)|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:
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| {{quote|There is just one point where I have encountered a difficulty. You state (p. 17 [p. 23 above]) that a function too, can act as the indeterminate element. This I formerly believed, but now this view seems doubtful to me because of the following contradiction. Let '''w''' be the predicate: to be a predicate that cannot be predicated of itself. Can '''w''' be predicated of itself? From each answer its opposite follows. Therefore we must conclude that '''w''' is not a predicate. Likewise there is no class (as a totality) of those classes which, each taken as a totality, do not belong to themselves. From this I conclude that under certain circumstances a definable collection [Menge] does not form a totality.<ref>Remarkably, this letter was unpublished until van Heijenoort 1967 – it appears with van Heijenoort's commentary at van Heijenoort 1967:124–125.</ref>}}
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| 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:<ref>Russell 1903:101</ref>
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| {{quote|Before taking leave of fundamental questions, it is necessary to examine more in detail the singular contradiction, already mentioned, with regard to predicates not predicable of themselves. ... I may mention that I was led to it in the endeavour to reconcile Cantor's proof...."}}
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| Russell wrote to Frege about the paradox just as Frege was preparing the second volume of his ''Grundgesetze der Arithmetik''.<ref>cf van Heijenoort's commentary before Frege's ''Letter to Russell'' in van Heijenoort 1967:126.</ref> 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,<ref>van Heijenoort's commentary, cf van Heijenoort 1967:126 ; Frege starts his analysis by this exceptionally honest comment : "Hardly anything more unfortunate can befall a scientific writer than to have one of the foundations of his edifice shaken after the work is finished. This was the position I was placed in by a letter of Mr Bertrand Russell, just when the printing of this volume was nearing its completion" (Appendix of ''Grundgesetze der Arithmetik, vol. II'', in ''The Frege Reader'', p.279, translation by Michael Beaney</ref> and proposed a solution that Russell would endorse in his ''Principles of Mathematics'',<ref>cf van Heijenoort's commentary, cf van Heijenoort 1967:126. The added text reads as follows: " ''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 function]]s that 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" (Russell 1903:522); The abbreviation Gg. stands for Frege's ''Grundgezetze der Arithmetik''. Begriffsschriftlich abgeleitet. Vol. I. Jena, 1893. Vol. II. 1903.</ref> but was later considered by some to be unsatisfactory.<ref>Livio states that "While Frege did make some desperate attempts to remedy his axiom system, he was unsuccessful. The conclusion appeared to be disastrous...." Livio 2009:188. But van Heijenoort in his commentary before Frege's (1902) ''Letter to Russell'' describes Frege's proposed "way out" in some detail – the matter has to do with the " 'transformation of the generalization of an equality into an equality of courses-of-values. For Frege a function is something incomplete, 'unsaturated' "; this seems to contradict the contemporary notion of a "function in extension"; see Frege's wording at page 128: "Incidentally, it seems to me that the expession 'a predicate is predicated of itself' is not exact. ...Therefore I would prefer to say that 'a concept is predicated of its own extension' [etc]". But he waffles at the end of his suggestion that a function-as-concept-in-extension can be written as predicated of its function. van Heijenoort cites Quine: "For a late and thorough study of Frege's "way out", see ''Quine 1955''": "On Frege's way out", ''Mind 64'', 145–159; reprinted in ''Quine 1955b'': ''Appendix. Completeness of quantification theory. Loewenheim's theorem'', enclosed as a pamphlet with part of the third printing (1955) of ''Quine 1950'' and incorporated in the revised edition (1959), 253—260" (cf REFERENCES in van Heijenoort 1967:649)</ref> For his part, Russell had his work at the printers and he added an appendix on the [[Type theory|doctrine of types]].<ref>Russell mentions this fact to Frege, cf van Heijenoort's commentary before Frege's (1902) ''Letter to Russell'' in van Heijenoort 1967:126</ref>
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| [[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")<ref>van Heijenoort's commentary before Zermelo (1908a) ''Investigations in the foundations of set theory I in van Heijenoort 1967:199</ref> laid claim to prior discovery of the [[antinomy]] in Cantor's naive set theory. He states: "And yet, even the elementary form that Russell<sup>9</sup> 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".<ref>van Heijenoort 1967:190–191. In the section before this he objects strenuously to the notion of [[impredicativity]] as defined by Poincaré (and soon to be taken by Russell, too, in his 1908 ''Mathematical logic as based on the theory of types'' cf van Heijenoort 1967:150–182).</ref> Footnote 9 is where he stakes his claim: | |
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| {{quote|<sup>9</sup>''1903'', pp. 366–368. I had, however, discovered this antinomy myself, independently of Russell, and had communicated it prior to 1903 to Professor Hilbert among others''.<ref>Ernst Zermelo (1908) ''A new proof of the possibility of a well-ordering'' in van Heijenoort 1967:183–198. Livio 2009:191 reports that Zermelo "discovered Russell's paradox independently as early as 1900"; Livio in turn cites Ewald 1996 and van Heijenoort 1967 (cf Livio 2009:268).</ref>}}
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| A written account of Zermelo's actual argument was discovered in the ''Nachlass'' of [[Edmund Husserl]].<ref>B. Rang and W. Thomas, "Zermelo's discovery of the 'Russell Paradox'", ''Historia Mathematica'', v. 8 n. 1, 1981, pp. 15–22. {{doi|10.1016/0315-0860(81)90002-1}}</ref>
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| 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".<ref>van Heijenoort 1967:124</ref>
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| In 1923, [[Ludwig Wittgenstein]] proposed to "dispose" of Russell's paradox as follows:
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| <blockquote> | |
| 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
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| 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)
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| </blockquote> | |
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| 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 [[type theory|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.
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| 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 [[Gödel's completeness theorem|complete]], [[Peano axioms|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.
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| 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.<ref>{{citation |url=http://books.google.com/?id=Xg6QpedPpcsC&pg=PA350 |title=One hundred years of Russell's paradox |author=Godehard Link |page=350 |year=2004 |isbn=978-3-11-017438-0}}</ref>
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| ==Applied versions==
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| 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.
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| As another example, consider five lists of encyclopedia entries within the same encyclopedia:
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| {| class="wikitable"
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| |- valign="top"
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| | width="20%" | List of articles about people:
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| *Ptolemy VII of Egypt
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| *Hermann Hesse
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| *Don Nix
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| *Don Knotts
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| *Nikola Tesla
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| *Sherlock Holmes
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| *Emperor Kōnin
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| | width="20%" | List of articles starting with the letter L:
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| *L
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| *L!VE TV
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| *L&H
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| *Leivonmäki
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| ... | |
| *List of articles starting with the letter K
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| *List of articles starting with the letter L ''(itself; OK)''
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| *List of articles starting with the letter M
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| ...
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| | width="20%" | List of articles about places:
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| *Leivonmäki
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| *Katase River
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| *Enoshima
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| | width="20%" | List of articles about Japan:
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| *Emperor Showa
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| *Katase River
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| *Enoshima
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| | width="20%" | List of all lists that do not contain themselves:
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| *List of articles about Japan
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| *List of articles about places
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| *List of articles about people
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| ...
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| *List of articles starting with the letter K
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| *List of articles starting with the letter M
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| ...
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| *'''List of all lists that do not contain themselves?'''
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| |}
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| 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.
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| 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".
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| 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.
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| One way that the paradox has been dramatised is as follows:
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| :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.
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| :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.
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| :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. | |
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| :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.
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| ==Applications and related topics==
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| === Russell-like paradoxes ===
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| As illustrated above for the Barber paradox, Russell's paradox is not hard to extend. Take:
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| * A [[transitive verb]] <V>, that can be applied to its [[substantive]] form.
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| Form the sentence:
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| :The <V>er that <V>s all (and only those) who don't <V> themselves,
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| Sometimes the "all" is replaced by "all <V>ers".
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| An example would be "paint":
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| :The ''paint''er that ''paint''s all (and only those) that don't ''paint'' themselves.
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| or "elect"
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| :The ''elect''or ([[Group representation|representative]]), that ''elect''s all that don't ''elect'' themselves.
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| Paradoxes that fall in this scheme include:
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| * [[Barber paradox|The barber with "shave"]].
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| * The original Russell's paradox with "contain": The container (Set) that contains all (containers) that don't contain themselves.
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| * The [[Grelling–Nelson paradox]] with "describer": The describer (word) that describes all words, that don't describe themselves.
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| * [[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''.)
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| == Related paradoxes ==
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| * The [[liar paradox]] and [[Epimenides paradox]], whose origins are ancient
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| * The [[Kleene–Rosser paradox]], showing that the original [[lambda calculus]] is inconsistent, by means of a self-negating statement
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| * [[Curry's paradox]] (named after [[Haskell Curry]]), which does not require [[negation]]
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| * The [[Interesting number paradox|smallest uninteresting integer]] paradox
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| == See also ==
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| *[[Cantor's diagonal argument]]
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| *"[[On Denoting]]", one of Russell's first attempts at critiquing Frege
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| *[[Self-reference]]
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| *[[Universal set]]
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| ==Notes==
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| {{reflist|2}}
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| ==References==
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| *{{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}}
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| *{{citation|last=van Heijenoort|first=Jean|authorlink=Jean van Heijenoort|date=1967, third printing 1976|title=From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931|publisher=[[Harvard University Press]]|publication-place=Cambridge, Massachusetts|isbn=0-674-32449-8}}
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| *{{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
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| }}
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| ==External links==
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| *{{MathWorld |title=Russell's Antinomy |id=RussellsAntinomy }}
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| *[http://www.cut-the-knot.org/selfreference/russell.shtml Russell's Paradox] at [[Cut-the-Knot]]
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| *[[Stanford Encyclopedia of Philosophy]]: "[http://plato.stanford.edu/entries/russell-paradox/ Russell's Paradox]" – by A. D. Irvine.
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| {{Set theory}}
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| {{DEFAULTSORT:Russell's Paradox}}
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| [[Category:Bertrand Russell]]
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| [[Category:Paradoxes of naive set theory]]
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| [[Category:1901 in science]]
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| [[Category:Self-referential paradoxes]]
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