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。


# by tnomura9 | 2016-12-09 02:16 | 考えるということ | Comments(0)

Semiotic set theory

Semiotic set theory

I think that it is easy to understand by handling the set semantically, because the intricacies of Russell's paradox in simple set theory seems to be easy to understand.

Semiotics is the concept of linguistics initiated by Saussure. The fundamental way of thinking is to consider the symbol as signifi- cant expression (signifiant) and symbolic content (sinifie) inseparably linked.

For example, the word "snow" is composed of the symbol "snow" and white crystals descending from the sky it points to. The term "snow" (symbol) is considered to be inseparably associated with the symbol itself (symbolic expression) and snow or concept (symbolic content) indicated by the symbol.

As the definition of a set in simple set theory is "a set is a collection of things", we can look at it as a semiotic point in the same way as the "snow". For the set A, the sign "set A" is consisted of the sign "A" itself as a symbolic expression and the group of things that set A is pointing to, that is, {a 1, a 2, ..., ak} as a symbolic content.

By considering a set as a semiotic symbol in this way it is possible to think separately from a set (symbolic expression) as a thing and a set (symbolic content) as a group of things that it points to.

Now, to simplify the discussion here, let's consider a collection {a 0, a 1, a 2, ..., an} consisting only of a set (symbolic expression). Then assume that each set points to the collection {ai, aj, ..., ak} of one of those sets (symbol contents).

In order to describe which element of this collection of sets (symbolic representation) points to which subset of the collection, we can create a n x n in table of the elements. In that table, the elements a 0, a 1, ..., an are arranged horizontally and vertically. Then In the row of the vertically aligned elements 1 is put when the element (set) contains an element corresponding to the horrizontally arranged a 0, a 1, a 2, ..., and 0 if not included as an element, at the intersection of the row and column of the table. In other words, the columns of 1 and 0 in the ai row indicate which kind of elements the set ai (symbolic representation) consists of (symbol contents).

** a 0, a 1, a 2, ..., an
a0 0, 1, 0, ..., 1
a1 1, 1, 0, ..., 0
a3 0, 0, 1, ..., 1

an 0, 0, 1, ..., 1

In this way, the correspondence relationship between the set ai (symbolic representation) and the group of elements (symbol contents) represented by the set ai can be described as a row of 1 and 0 in line ai. In other words, whether set ai contains set ak as an element can be determined by whether or not the number of ak columns in line ai is 1.

After that, examine the diagonal part of this table. 1 or 0 described there indicates whether the set contains itself as an element or not. For example, if the value of ai row ai column is 1, set ai includes itself as an element, and if the value of ak row ak column is 0, set ak does not include itself as an element.

Therefore, if you collect the sets ak, ..., am whose diagonal values ​​are 0, you can create sets of sets that do not contain itself as an element. Let's call it ar (symbolic expression) for borrowing. What happens to the line of 1 and 0 of the ar row corresponding to the set {ak, ..., am} represented by ar (symbolic representation)? Clearly it turns out that if the diagonal value of the table is 0, it is 1 and if 1 is 0.

Well, what matches the sequence of this ar line is in a 0, ..., an rows in the table? Obviously it is impossible. Because the sequence of ar rows is the inverted value of the diagonal, so it is different for every diagonal part of any ai.

Therefore, even though there is a group of sets that do not contain themselves as elements in the set {a 0, ..., an}, the symbolic expression of it can not exist in the table which describes the correspondence of the sets and its contents.

For this reason, despite the fact that there is a group of sets that do not make themselves an element, a set ar whose symbolic content is a symbol can not be found in a 0, ..., an. In other words, there can not be a symbolic expression named set ar with those gatherings as symbolic contents. If you are going to instruct such a gathering, it should be called a class, not a set. In Russell's paradox, the paradox was deduced because we thought of a set that do not exist like this as a set.

In this way, by considering a set semiotically as a symbol composed of symbolic representations (signifiers) and symbolic contents (signifiers), it is important to clarify the reasons for classes such as those found in Russell's paradox it can.

From the above discussion, the set of Russell's "set of sets that do not contain itself as an element" can not be a set in any case, and that group will become a class, but the classes includes Russell's class are not there anything else? The answer is understood by thinking as follows.

Obviously, there are actually 2 ^ n collections whose elements are set a 0, a 1, ... an as a symbolic representation. However, there are only n sets that can be considered as symbolic expressions. All other gatherings can not be represented as a set, and they all become classes. The difference between the number of sets and the number of classes increases as the set of symbolic expressions increases. There will be far more classes at any point, even if you set the set as a symbolic expression infinitely large.

In this way, it is easier to understand the origin of the Russell's paradox and the existence of the class by grasping the set semiotically, that is, consider a set as a composition of a symbolic expression and the extension as its symbolic contents.

いきなり英文の記事で不審に思われたかもしれないが、過去記事の「記号論的集合論」を Google 翻訳にかけてみただけだ。明らかに変なところは手直ししてみたが Japanish になってしまったかもしれない。


英文で論文やマニュアルを書いたり、単に英作文の勉強をするのにも Google 翻訳は便利なのではないだろうか。人工知能には期待できそうだ。

# by tnomura9 | 2016-12-05 03:25 | 考えるということ | Comments(0)

Googld 翻訳の実力

Neural Networks and Deep Learning

の扉の文章を Google 翻訳で訳してみた。

Neural Networks and Deep Learning is a free online book. The book will teach you about:

Neural networks, a beautiful biologically-inspired programming paradigm which enables a computer to learn from observational data
Deep learning, a powerful set of techniques for learning in neural networks

Neural networks and deep learning currently provide the best solutions to many problems in image recognition, speech recognition, and natural language processing. This book will teach you many of the core concepts behind neural networks and deep learning.

For more details about the approach taken in the book, see here. Or you can jump directly to Chapter 1 and get started.

ニューラルネットワークとディープラーニングは無料のオンラインブックです。 この本は次のことをあなたに教えるでしょう:


ニューラルネットワークと深層学習は、現在、画像認識、音声認識、および自然言語処理における多くの問題に対する最良の解決策を提供しています。 この本は、ニューラルネットワークと深層学習の背後にある多くの核心概念を教えてくれるでしょう。

本書で取り上げているアプローチの詳細については、こちらを参照してください。 あるいは、第1章に直接ジャンプして開始することもできます。


# by tnomura9 | 2016-12-03 14:13 | 考えるということ | Comments(0)

Newral networks and deep learning

Michael Nielsen 氏の 2016 年のネットブック

Neural networks and deep learning



# by tnomura9 | 2016-12-02 19:36 | 考えるということ | Comments(0)


elllo はフリーの英語のリスニング・レッスンサイトだ。サイトの案内の文章をグーグル翻訳で翻訳してみた。

What is ELLLO?
ELLLO! Welcome to English Language Listening Library Online. My name is Todd Beuckens and I am an ESL teacher in Japan. I created ELLLO to help students (and teachers) get free listening lessons online. I post two new lessons each week. Get free lessons and updates via email here .

エルロ! オンライン英会話リスニングライブラリーへようこそ。 私の名前はTodd Beuckensで、私は日本のESL教師です。 私はELLLOを生徒と教師が無料のリスニングレッスンをオンラインで手助けできるように作成しました。 私は毎週2つの新しいレッスンを投稿します。 無料のレッスンや最新情報をEメールで入手してください。







TED もおすすめだ。

# by tnomura9 | 2016-12-02 00:35 | 考えるということ | Comments(0)


人工知能の解説ビデオをみていたらシンギュラリティという言葉が頻繁に出てきたので調べてみた。技術的特異点 (Singularity) というらしい。人間が人間の知性を超える AI を生み出した時点で AI が更に優れた AI を作りはじめ、人間が到達できない超知性が生まれるというものだ。



SF じみて怖い話だが、人工知能をよく知る人達の間で真面目に議論されているらしい。

人工知能で驚異的なのはその学習速度だ。Google の AlphaGo は開発が始まったのが 2014 年で、2016 年 3 月には囲碁の世界チャンピオンを破っている。人間のそれも選ばれた資質の棋士が半生をかけて集中して学習して得た知識のレベルに達するのに2年しかかかってない。

それは囲碁というゲームだからできたことだと考えるかもしれないが、チェスで人間に勝利した IBM のディープブルーは汎用人工知能のワトソンに発展し、人間の医師で解決できなかった血液疾患の診断と治療を提案し、治療を成功させるところまで行っている。2016年にして人類は自分たちより学習能力の優れた人工知能と共存することを学び始めなくてはならなくなっているのだ。


# by tnomura9 | 2016-11-27 23:11 | 考えるということ | Comments(0)

汎用人工知能 H












# by tnomura9 | 2016-11-27 01:38 | 考えるということ | Comments(0)






# by tnomura9 | 2016-11-25 08:21 | 考えるということ | Comments(0)


AI を活用しようと思ったら、ネットの情報を自動で収集する方法を知らないといけないと思った。情報収集を自動でできたら、AI が勝手に情報を集めてくれて、勝手にデータベースを構築してくれる。また、情報を電子化しないと AI に活用できない。



# by tnomura9 | 2016-11-24 04:11 | 考えるということ | Comments(0)

オープンソースの AI ライブラリ

AI の威力を少し脅威に感じたのでオープンソースで使えるものがないかどうかを探したら、Google の TensorFloor と Preffered Networks 社の Chainer をインストールして動作チェックまでやった記事を見つけた。

どちらも、C++ か Python で動かすらしい。管理人は残念ながらどちらも使えない。Pytnon でも勉強してみようかな。

この記事を書いているときに池上彰の番組で AI の取材映像を放送していたが、学習速度が驚異的で、学習の成果を他のロボットに移植できて、さらに、24時間疲れ知らずに働き、文句も言わない。どの点をとっても人間はかなわない。

AI 時代の生き残りについて真剣に考える必要がある。

# by tnomura9 | 2016-11-23 22:23 | 考えるということ | Comments(0)