Radix

In this item, I explain a radix in the mathematics. Please see "a radix" (vagueness evasion) about other uses.

References and the source which I can inspect are not shown at all, or this article is insufficient. You add the source, and please cooperate with the reliability improvement of the article. (February, 2015)

Aleph zero, the smallest infinite radix

The radix (cardinal number to come, and to suck in or cardinals) is generalization of the natural number as the thing to measure カーディナリティ (density, size, size) of the meeting in mathematics. The number of the element of density (cardinality) that is the limited set of the limited meeting is natural number. The size of the infinite meeting is described in 超限基数.

The density is defined using bijection. It means that bijection exists between the meetings to have the density that two meetings are equal in. In the case of a limited meeting, I may agree to an intuitive concept of the size. In the case of an infinite meeting, the behavior is complicated. The density of infinite meeting was not only one kind, and the basic theory that Georg Cantor showed showed a thing. Particularly, the density of set of real number showed that it was big truly than the density of set of the natural number (theorem of Cantor). In addition, it is possible that the density of the truth subset of the limited meeting and the truth subset of the meeting infinite for the density of the original meeting not being able to equal is equal to the density of the original meeting (it refers to the Dede quinte infinity).

超限列 of the radix exists:

${\displaystyle 0,1,2,3,\ldots ,n,\ldots ;\aleph _{0},\aleph _{1},\aleph _{2},\ldots ,\aleph _{\alpha },\ldots .\}$

The natural number that is a limited radix forms a line first, and, in this row, the aleph number (aleph number) that is the infinite radix of the well-ordered set continues afterwards. The aleph number is added an additional character to by ordinal number. Under the supposition of the axiom of choice, this 超限列 includes all radices. The situation is complicated about the infinite radix that is not the aleph number more if axiom of choice is not assumed.

The density is studied for a part of set theory. In addition, I am used as a tool of each field of the mathematics including a combination theory and abstract algebra, analytics. The radix forms skelton of the area of the meeting of the area idea.

Table of contents

History

The concept of the density was formulated by Georg Cantor who was the founder of set theory. The density is used to compare one side of the limited meeting. For example, {4, 5, the meeting of 6} are not equal to {1, 2, 3}. However, ({1,->4, 2->5, 3->I have the same "density" of existence) 3 that, thus, I was established of one-on-one correspondence of 6}.

A concept that Cantor supports one to one, for example, meeting N = {0, 1, 2, 3 of the whole natural number, ...I applied to an infinite meeting such as}. Radix same by the meeting that there is one-on-one correspondence between N as for the countable infinity meeting and the good denumerability infinity meeting${\displaystyle \aleph _{0}}$ I have (aleph zero). Cantor called the radix corresponding to such an infinity meeting 超限基数.

Cantor might be against intuition, but any me on world subset of N proved that I had density same as N, too. In addition, the order of natural number vs. the whole proved that it was countable infinity, too (this leads that a set of the whole rational number is countable infinity promptly). In addition, the set of the whole algebraic figure proved that it was countable infinity later, too.

Cantor showed that the radix of the high rank existed by showing that the density of set of the whole real number was bigger than the density of N truly in an article of 1874. His proof was complicated logic using the section reduction method. However, I was full of invention and, in the article of 1891, proved the same thing using what's called concise diagonal logic. The new radix corresponding to the set of the whole real number is called continuum density; Cantor${\displaystyle {\mathfrak {c}}}$ I used という sign for it.

Cantor developed general theoretical most of the radices. He showed existence of smallest 超限基数. In addition, about any radix, I showed that a big radix existed next.

His continuum hypothesis,${\displaystyle {\mathfrak {c}}}$  は ${\displaystyle \aleph _{1}}$  It is the proposition that に is equal to. The continuum hypothesis is the meaning that neither the proof nor the negation can prove from axiom system, and an independent thing is shown from normal axiom system of set theory (axiom system of Zell Melo Frenquel).

Motive

The correction of this knob is expected.

Official definition

I define it as the density of meeting X among α which there is bijection between X and ordinal number α if I assume axiom of choice formally when smallest. This definition is known as a radix problem (en:Von Neumann cardinal assignment) of von Neumann. There is not it if I do not assume axiom of choice if it is in different this year and rejects it. (was shown by Cantor definitely indirectly in Frege and プリンキピア マテマティカ) the oldest definition of the density of meeting X of X and the のつくであるすべての meeting for one to one [is a definition as class ][X]. This does not function in ZFC and the axiom system of associated set theory well. If X is non-empty, it is because the thing which collected のつくであるすべての meetings for one to one is too big for a meeting. Actually, injections from the space to [X] exist by thinking about representation from meeting m to {m} X X about meeting X which is not empty, and [X] is a true class than a limit (en:Limitation of size) of the size.

Radix operation

The correction of this knob is expected.

Continuum hypothesis

The details refer to "a continuum hypothesis"

With the continuum hypothesis, it is aleph-null ${\displaystyle \aleph _{0}}$  と continuum density ${\displaystyle 2^{\aleph _{0}}}$  I point to the proposition that there is not a radix in the の interval. In other words, ${\displaystyle 2^{\aleph _{0}}=\aleph _{1}}$  But, it means that it is managed. In addition, with the general continuum hypothesis, it is | for any meeting X X | と 2^| X | It is the proposition that there is not a radix in the の interval. With both propositions, it is proved that it is independent by ZFC.

Allied item

• Huge radix
• Inaccessible radix

Footnote

It is acquired by "https://ja.wikipedia.org/w/index.php?title= radix &oldid=61661013"

This article is taken from the Japanese Wikipedia Radix

This article is distributed by cc-by-sa or GFDL license in accordance with the provisions of Wikipedia.

Wikipedia and Tranpedia does not guarantee the accuracy of this document. See our disclaimer for more information.

In addition, Tranpedia is simply not responsible for any show is only by translating the writings of foreign licenses that are compatible with CC-BY-SA license information.