Russell's Paradox

1. Russell's Paradox

The paradox of Russell is a paradox about naive set theory discovered by UK philosopher and mathematician Bertrand Russell. that is,

"The set of all sets that are not members of themselves can neither be a member of itself nor be not."

Consider this set as R, if R does not contain itself, R must be a member of R as R is "a set that is not a member of itself". On the other hand, if R contains R as an element, this must satisfy the intensional definition "a set that is not a member of itself", so R must not contain R as an element. In either case a paradox will occur.

Since sets should be divided into either sets that contain themselves or sets that do not contain themselves, if a paradox occurs in one of them, the method of defining a set with an intensional definition can not be used. Furthermore, since most of the sets seems not to contain themselves, the occurrence of a paradox in the collection of them will shake the foundation of set theory.

By definition that "a set" is considered as "an object" of collections of objects. And according to the comprehension axiom, a set can be defined by collecting objects that satisfy the predicate. But, the unmistakable paradox is hidden under these definition. You can imagine the hurdles of logic scholars and mathematicians. Even so, what is the secret mechanism of the paradox?

2. Mechanism of Russell's paradox

Let's see what mechanism causes Russell's paradox.

Russell's set is "a collection of all the sets that do not contain themselves". But before collect all of such sets, let's consider collect a suitable number of "sets that do not contain themselves" rather than all, and name the set R ' . For example, "the collection of dogs" is not a member of the collection of dogs, therefore, "the set of dogs" is "a set that does not contain itself". You can easily compose R' with those sets like "a set of dogs" if you do not consider collection of all of them.

And then, let us consider whether R ' itself is contained R'. From the intensional definition it is clear that R ' is not a member of R'. This is because if R' is contained in R' as an element, R' as an element becomes a set that contains itself. Therefore, R' never contains R' itself. However, for that matter, R' satisfies the predicate "a set that does not contain itself".

In the case of such a set, however, it is not a problem because the intensional definition is a collection of "suitable number of the sets which do not contain themselves". This set can exist stably. However, no matter what kind of R' is made, R' satisfies the predicate "a set that does not contain itself" in spite of R' does not contain R'. Therefore, we can not make the set that collects ALL the sets that do not contain themselves such as Russell's set. That means Russell's set cannot exist as "a set". Such collection should be thought as "a class".

In Russell's paradox it happened because such a collection was thought as a set. From that point of view, it is not permissible to have a collection that collects sets that do not contain themselves and not a member of itself at the same time. However, paradoxes will not occur if such a collection is classified as a class and distinguished from a set.

The point is, considering a set as an object is the essential cause of paradox. If you think about a set is an object, collection of objects it denotes must be distinguished from itself. In Saussure's semiotics, the sign is the inseparable combination of the signifier and the signified, and a set itself is apparently a signifier and its extension is a signified. From this point of view, you can easily imagine what is a set that contain itself and what is a set that does not contain itself. Both case can be occur in that model, although, any set which collect the sets that does not contain themselves never be a member of itself.


前回の記事で Google 翻訳で英作文したのが面白かったのでもう1回やってみた。元の原稿は過去記事の「ラッセルのパラドックスの謎が解けた」から取ったが、最後の結論の部分は Google 翻訳に対話的に日本語の文章を入力しながら作成した。その際、意味のある英文が出てくるまで日本語の文章の方を修正した。日本語の文章を短い明確な表現に変えると、翻訳された文章も英文らしい表現になっていくのが面白かった。

アイディアを英文でアップすれば、海外の人も読んでくれるかもしれない。機械翻訳が発展すれば、そんな夢のような時代がやってくるのだろう。

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by tnomura9 | 2016-12-09 02:16 | ラッセルのパラドックス | Comments(0)
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