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 | ラッセルのパラドックス | Comments(0)
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